MAU B WN RP
. Crystal Growth-Bulk and Epitaxial Film-Part 1
. Crystal Growth-Bulk and Epitaxial Film-Part 2
. Crystal Growth-Bulk and Epitaxial Film-Part 3
. Crystal Growth-Bulk and Epitaxial Film-Part 4
. Tutorial on Chapter 1-Crystal and Crystal Growth.
. Syllabus of EC1419A_ Electrical and Electronic Materials
Science
. Chapter 2. Solid State of Matter.
. Chapter 2. Section 2.3.2. Application of Inert Gases.
. Chapter 2. Solid State of Matter. Section.2.5. Bohr Model of
Hydrogen Atom.
. Tutorial on Chapter 2: Insulator, Semi-conductor and Metal.
. SSPD_ Chapter 1_ Part 12 Quantum Mechanical Interpretation
of Resistance
. SSPD Chapter 1 Part 11 Solid State of Matter
. Chapter 3. Special Classification of
Semiconductors.Sec3.1.Compund Semiconductors
. Chapter 4.Light and Matter — Dielectric Behaviour of Matter
. Section 4.2. Dielectric and its Physics.
. Section 4.3. Dielectric Loss.
. Section 4.5. Piezo-electricity and Ferro-electricity.
. Chapter 5_Introduction-Nanomaterials.
. Section 5.4. Nano-Size Effect on various opto-electronic-
magnetic properties of nano materials.
. EVEN SEMESTER2014-MidSemester Examination Answers
. Chapter 5. Section 5.6. Metamaterials.
. Chapter 7. Magnetic Materials.
. Chapter 7. Section 7.6.3. Hysteresis Loops of Hard Iron and
Soft Iron.
. Chapter 8. Superconductors.
Crystal Growth-Bulk and Epitaxial Film-Part 1
Crystal Growth Part 1 goes on to describe the preparation of electronic
grade polycrystalline Si from sand and then preparing single crystal pure or
doped Silicon Ingot with a given crystalline orientation.
CRYSTAL GROWTH
e (1)Preparation of polycrystalline Electronic Grade Silicon material in
Siemens Reactor.
e (2)Bulk Crystal preparation or Pure,Electronic Grade, Single Crystal
Silicon ingot preparation by Czochralski method and by Float Zone
Method.
e (3)Diamond cutting of Silicon Ingot into Silicon Wafers and lapping,
polishing and chemical etching of the Wafers to obtain mirror finished
Substrates for IC preparation.
e (4)Epitaxial Film Growth-Chemical Vapour Phase Deposition(CVD)
or Liquid phase Epitaxy(LPE).
e (5)For Photonic Application , compound Semiconductor Epitaxial
Film Growth is achieved by Molecular Beam Epitaxy (MBE).
PREPARATION OF POLYCRYSTALLINE ELECTRONIC GRADE
SILICON MATERIAL.
Quartz Sand (SiO2) plus Coke are put in a container with submerged
electrode Arc Furnace. Electric energy is consumed at the rate of 13 kWhr
per kg and SiO2 is reduced to Silicon by Coke. This silicon is of 98%
purity. The solid silicon is pulverized and kept in an oven where Hydrogen
Chloride vapour is passed at 300°C. Silicon is converted into SiHCI3
(liquid) which is called Trichloro Silane.
SiO2(Quartz Sand)+ 2C (coke) = (at high Temperature)Si +CO2
Si(pulverized form) +3HCl (vapour) =(At 300°C) SiHCI3 (liquid) +H21
°o
0
SS
Ss Pure SiHC13
pees
Figure 1. Multiple Distillation Set-up for preparing electronic grade
TriChloroSilane.
SiHCI13(liquid) has B.P 31.8°C. Whereas most impurities are less volatile.
Therefore by multiple distillation electronic grade TriChloroSilane is
obtained which eventually is used in SIEMEN’S REACTOR.
H2 + SiHCL3 =(at 1000°C) Si +3HCL
(Hydrogen reduction)
(a form of CVD of Silicon on slim rod of Silicon on Tantulum wire)
Slim rod contains Tantalum Wire surrounded by Silicon as shown in Figure
2. Slim rod can be formed either by CVD on hot Tantalum wires or can be
pulled from the melt.
(When Si deposited by CVD on hot Tantalum wires then there can be
thermal run away due to negative temperature coefficient of resistance(t.c.r)
of surrounding silicon deposit dominating over positive t.c.r of the
Tentalum wires. The thermal runaway has to be prevented by controlling
the current.)
The resulting fattened slim rod has considerable metal contamination at the
core of the rod. This can be removed by dissolving the central metallic core
with acid. ANitric/Hydrofluoric acid mixture is then pumped through the
hollow core. This cleaning method widens the hollow core by removing
silicon and metal contamination(1mm/hr). First nitric acid is used to
dissolve the metallic core . Next Hydrofluoric acid is passed through the
hollowed core to dissolve the inner layer of silicon and in the process
removing the metal contamination. The fattened, purified rod of silicon thus
obtained will be used as the feedstock for crystal pulling. Hydrofluoric acid
is very corrosive and it can effect our bones even . Therefore Teflon gloves ,
Teflon Aprons, Teflon tweezers and Teflon beakers have tobe used while
working with HF acid.
Transparent Glass Dome
CVD of Si on Slim Rod with tantulum Core
SIHCI3 +H2
aN
Aces
Exhaust HCl
Vapour.
Figure 2. Siemen's Reactor for Chemical Vapour Deposition of Electronic Grade
Si on Tantulum Wire Core which is heated at 1000 degree centigrade.
FLOW CHART OF SIEMEN’S REACTOR
Reduction of sand with carbon gives impure polycrystalline Silicon
1
Reaction of pulverized raw silicon with HCI gaseous vapour to form
TriChloroSilane
{
Multiple distillation of TriChloroSilane to obtain purified electronic grade
TriChloroSilane
Thermal decomposition of SiHCI3 at 1000 degree centigrade in Sieman’s
reactor to obtain fattened rods of electronic grade Silicon
Pull shaft
Rotation |) Inert gas
Inert gas in i)
i | <—Seed Crystal
we Ingot
Growing
Crystal
Quartz container—-
Quartz crucible
RF induction © eel re & RF
coll to heat @ -— i @ Induction
graphite rT Molten me Coil
susceptor & Silicon =] &
|_| |
& =
@ _ | &
fe Se
Graphite Susceptor [ : J
Inert Gas out Y
Figure 3. The schematic illustration of the growth of a single crystal Si
Ingot by Czochralski Technique.
Figure 3. The schematic illustration of the growth of a single crystal Si
ingot by the Czochralski technique.
(100) plane
Single crystal Si ingot( about 2m) Cut wafer
[100] Direction
Figure 4.The crystallographic orientation of the Silicon Ingot is marked
by grounding a Flat. The ingot can be as long as 2 meter. Wafers are
cut using a roating annular diamond saw. Typical wafer thicknes is
0.6-0.7 min.
The crystallographic orientation of silicon ingot is marked by grounding a
flat as shown in Figure 4. The ingot can be as long as 2m. Wafers are cut
using a rotating annular diamond saw. Typical wafer thickness is 0.6-0.7mm
CZOCHRALSKI METHOD OF SINGLE CRYSTAL PULLING
Polycrystalline silicon is not suitable for Electronic Device preparation.
Polycrystals have grain boundaries which make mobility unpredictable
because of unpredictable amount of defect scattering. Grains boundaries
cause uneven distribution of dopents. Along the grain boundaries there is
more rapid diffusion of dopent as compared to that in the bulk. Hence we
opt for single crystal silicon
Czochralski Equipment, as shown in Figure 3 and Figure 5, has a graphite
lining. Graphite is of Nuclear Reactor Grade. Within the Graphite lining,
Quartz crucible of high purity is placed for holding the silicon melt.
Induction heating is used for maintaining the silicon melt at 1420°C .
Induction heating is eddy current heating. The eddy currents are induced in
the graphite lining by alternating magnetic field caused by RF induction
coil. Crystal pulling is carried out in inert atmosphere of argon to prevent
oxidation and to suppress evaporation.
Rotating Support Rod
Single Crystal
/ being pulled
Quartz Crucible
(Refractory Material)
& RF Induction
@& Heating Coil
Seed Crystal
Crystal Pulling
Direction
~ Silicon Melt”
1420 deg. cent.
Counter Rotating Susceptor
Support
Figure 5. Czochraski Equipment.
Seed crystal determines the perfection and the orientation of the single
crystal being pulled. Hence seed is dislocation free and of desired crystal
orientation.
The total length of Si ingot is L=2m & ® =15cm, 20cm, 25cm, Optimum
pulling rate is 2 mm per min. In 24 hr period the total length of ingot is
pulled .Silicon being highly refractive (m.p.1410 c) is highly contaminable.
The Quartz of the containment vessel undergoes dissolution into Si during
its growth. The result is the inclusion of O2 as a donor impurity into
Silicon. Carbon is another impurity that finds its way into Silicon. These
contaminants are particularly harmful for powerful devices.
Counter rotation and rotation are produced to provide uniformity ,
homogeneity of thermal effects and dopents. This minimizes the defects
also.
Inspite of these homogenizing effects there are swirl effect which lead to
non-uniform distribution of desired and undesired contaminants. This leads
to non-uniform resistivity distribution across a silicon wafer. The spatial
variation is such that these non-uniformities are not particularly important
in I.C. or in miniature devices. However in power devices ,which are large
area devices , local variation in doping can lead to non-uniform heating and
hence to hot spots and failures. For example 3” diameter Thyristor carries
3000 amps. Even one defect per slice will be catastrophic to such a device
but is inconsequential to I.C. Fabrication.
This implies that for applications in which contaminants are critical, crystal,
wafers produced by CZLOCHRALSKI METHOD are not suitable.
One way to avoid the contaminants is to be avoid contact with quartz
crucible. This is achieved by FLOAT ZONE METHOD of Crystal Growth.
FLOAT ZONE METHOD OF CRYSTAL GROWTH
An electronic grade , pure polycrystalline silicon rod is held vertically as
shown in Figure 6 in inert atmosphere .thus no crucible is used and hence
contaminations are avoided.
A seed crystal is held at the bottom of the rod to determine the
crystallographic orientation
Through R.F induction heating a narrow zone of vertical, purified, poly
crystalline bar is melted. Dimensions of the molten zone are such that
surface tension hold it in place. Because of less than unity segregation
coefficient , a(m)=(concentration M in solid phase)/(concentration of m in
liquid phase), impurities prefer to stay in liquid phase.
As the R.F. coil is sifted from bottom to top , the impurities of the region
are swept to the top and the recrystallized portion is relatively pure and
single crystalline with orientation that of seed crystal. If high purity is
required then several passes can be made and ends containing impurities
can be sliced off leaving behind very pure single crystal silicon ingot. Pure
crystal of very high resistivity can be achieved by repeated recrystallization
through several passes.
Table 1. Segregation Coefficient for different dopents in different hosts.
Semiconductor N Type P type
Silicon a(P) = 0.35, a(As) = 0.3 a(B) = 0.8
Gallium Arsenide a(Se) = 0.10 a(Zn) = 0.42
Rotation
—_—>
Argon Inlet
Cooled Direction of
Silica Envelope travel of
RF Coil
Molten
Silicon RF Coil
maintained by
eddy current heating Single
Crystal
Seed
Counter-Rotation Argon Outlet
——
Figure 6. Float Zone Method of Crystal Growth.
Crystal Growth-Bulk and Epitaxial Film-Part 2
Part 2 of Crystal Growth describes the sawing of the Ingot into taper free,
flat Si Wafers. Lapping , chemical etching and polishing are described to
obtain mirror finish, defect free 300mm diameter Silicon Wafer.
Crystal Growth- Bulk and Epitaxial Film- Part2.
SLICING OF SINGLE CRYSTAL INGOT INTO SILICON WAFERS
Slicing of the Si ingot into Si wafers is achieved by circular saw blade. The
circular saw blade is illustrated in Figure 7.
Stepped Forward i
in equal steps
Longitudinal axis of Si Ingot
Figure 7. Circular Saw Blade with diamond
lmpregnation. Saw Blade must be kept perpendicular
to the lontitudinal axis of the ingot to avoid taper in
the wafer.
Circular saw blade consists of stainless steel impregnated with diamond
dust at the cutting edge. On the hardness scale diamond is the hardest. 500
micrometres thick Si wafers are sliced out through damage free and parallel
sawing. The ingot should be sawed in parallel planes to avoid taper of the
wafer.
Wafer
Non-uniform thickness of the wafer gives
rise to TAPER defect
| |
By BOW or CURVATURE of Wafer ,
flatness is effected.
Figure 8. TAPER & FLATNESS of Wafer
In the process of sawing , mechanical, defects are inevitable. To remove the
mechanical defects and to achieve taper-free, flat, mirror finish wafers we
do lapping, chemical etching and polishing.
LAPPING: For removing mechanical scratches and abrasions encountered
during sawing, lapping is essential. A batch of wafers are placed between
two parallel plain steel discs which describe planetry motion in relation to
each other. A slurry of cutting compound (generally alumina) is used and as
lapping progresses, the grade of abrasion is refined. Smoothness is within
10 um and flatness is within 2 pm.
CHEMICAL ETICHING: To remove edge damages which had been
caused due to sawing and grinding, the wafers is dipped in HF acid using
Teflon beakers. Chemical Vapor etching can also be done as described in
epitaxy section.
POLISHING: A slurry of silica in NaOH is used for polishing to mirror
smoothness. Lapping , Chemical Etching and Polishing removes 100pm
thick substrate leaving behind 400um thick silicon substrate on which I.C.
Fabrication can be carried out..
FLOW CHART OF WAFER PREPARATION
Starting Material - Sand, Sand reduced to Silicon, Silicon converted to
TriChloroSilane, through multiple distillation TrichloroSilane is purified to
Electronic Grade and reduced in Siemen’s Reactor to Pure Electronic Grade
Polycrystalline Si.
{
From purified Silicon Melt, single crystal ingot is pulled out by Czochralski
Method or Poly Si ingot is purified re- crystallized by Float Zone Method.
Required Dopent ( Phosphoprous for N Type ingot and Boron for P Type
ingot) is added to the melt in calculated manner to form single crystal ingot
of specific impurity type doping and of given resistivity.
ui
Sawing, Lapping, Grinding and Polishing
{
Polished Si Wafer
Diameter of Wafer
200min in 80's & 90's
Today it is 300min.
In future it will be 400mm
to maintain the scale of economy
| Thickness of the Wafer
400microns
In early days when IC Technology had just started i.e. in the year 1961 the
wafer size was only 1 inch. Then it was: * 2 inch (50.8 mm). Thickness 275
pm. * 3 inch (76.2 mm). Thickness 375 pm. * 4 inch (100 mm). Thickness
5925 pm. * 5 inch (127 mm) or 125 mm (4.9 inch). Thickness 625 pm. * 150
mm (5.9 inch, usually referred to as "6 inch"). Thickness 675 pm. * 200
mm (7.9 inch, usually referred to as "8 inch"). Thickness 725 pm. * 300
mm (11.8 inch, usually referred to as "12 inch" or "Pizza size" wafer).
Thickness 775 pm. * 450 mm ("18 inch"). Thickness 925 jam (expected).
By all these technique , high purity (99.999%) single crystal ingots with a
controlled amount of N or P type dopent can be produced. The maximum
resistivity obtained in Si ingot is 10* Q-cm whereas in Ga As and InP the
pmax =10°Q-cm. That is almost semi-insulating ingot of GaAs and InP can
be obtained. Semi-insulating GaAS and InP wafers are suitable for
reduction of parasitics, for isolaton of active devices from one another, for
reduction of power drain and for achieving high speed.
Presently because of technical constraints we have the option of Si, GaAs
and InP substrate only. Therefore we have a very limited choice of overlay
films also. This severely constrains the photonic device design.
Crystal Growth-Bulk and Epitaxial Film-Part 3
Part 3 of crystal growth sums up the properties of Silicon, describes the
identity marks of Silicon Wafers in terms of Semiconductor Type and in
terms of Crystal Orientation and this part defines the crystal orientation and
its relevance in IC fabrication.
CRYSTAL GROWTH
Silicon, (Si): The most common semiconductor, atomic number 14,
energy gap Eg= 1.12 eV- indirect bandgap;
crystal structure- diamond, lattice constant 0.543 nm,
atomic concentration 5 x 10** atoms/cm’,
index of refraction 3.42, density 2.33 g/cm?, dielectric constant 11.7,
intrinsic carrier concentration 1.02 x 10/9 cm’,
bulk mobility of electrons and holes at 300°K: 1450 and 500 cm?/V-s,
thermal conductivity 1.31 W/cm°C,
thermal expansion coefficient 2.6 x 10°°°C"t,
melting point 1414°C; excellent mechanical properties (MEMS
applications);
single crystal Si can be processed into wafers up to 300mm in diameter.
In future this diameter will be 450mm.
P type= Always Boron (B) Doped N type= Dopant typically as follows:Res:
.001-.005 Arsenic (As)Res: .005-.025 Antimony (Sb)Res: >.1 Phosphorous
(P)
EPITAXIAL CRYSTAL GROWTH
The substrate or the wafer only constitutes the strong base of the integrated
circuit. The actual active and passive components fabrication and there
integration are carried out in overlay films which are grown by epitaxial
technique.
EPITAXY is a Greek word meaning : ‘epi’ (upon) & ‘taxy’ (ordered). That
is an epitaxial film, a few pm thick, is an orderly continuation of the
substrate crystal. It grows very slowly layer by layer. Hence the dimension ,
defects and doping magnitude as well as uniformity can be precisely and
accurately controlled in the crystal growth direction.
This precise control is obtained in Molecular Beam Epitaxy (MBE) but not
in Liquid Phase Epitaxy(LPE) or in Chemical Vapour Phase Epitaxy
(CVPE). The thickness accuracy is within +3A which is essential for
growing Quantum Photonic Devices namely Quantum Dots, Quantum
Wells and Super-lattices
Primary Flat
Secondary Flat
Figure 9 Silicon Wafer Identity Marks.
Table 2. Identity marks of the Wafer to identify its orientation and
semiconductor type.
a (angle between primary and secondary
flats as indicated in Figure 9) Type Orientation
45° N <111>
90° P <100>
180° N <100>
0° P <111>
The normal to the plane along which crystals cleaves is the cleavage plane
orientation. Suppose the cleavage plane orientation is <111>. Miller Index
is being used to define the planes and their normal. Figure 10 illustrates the
Plane’s Miller Index and how the normal to the plane is represented. If the
exposed surface of the Si wafer, which is known as major flat, is parallel to
cleavage plane then the given wafer has a crystal orientation <111>.
If the cleavage plane orientation is <100> and the wafer major flat is
parallel to YZ plane then the crystal orientation is <100>. In this case
cleavage plane lies in YZ plane i.e. [100] plane and its orientation is
perpendicular to YZ plane i.e. x-axis. Hence Wafer Crystal Plane
orientation is <100>
Scribing the wafer along cleavage planes allows it to be easily diced into
individual chips (‘die”) so that billions of individual circuits or systems on
an average wafer can be separated into individual dies. Each individual die
is eutectic ally bonded on ceramic substrate. The substrate is bonded to the
header.The gold wire is connected to the bonding pads of the die on one end
and to the chip terminals on the header by Thermo-compression bonding or
by Ultra-sonic bonding. Next the die is hermetically sealed into Dual-in-
Line(DIP) package or TO5 package
In <100> crystal orientation, scribed pieces form rectangle whereas in
<111> crystal orientation, scribed pieces form triangles. Here we have to
scribe from the base of the triangle to the apex.
For MOS fabrications, wafers with crystal orientation <100> are used. This
helps achieve a lower threshold voltage. For BJT and other applications
wafers with orientation <111> are preferred.
Silicon Crystal Bulk is isotropic to diffusion of dopents and to etchents used
for etching the oxide layer. This is because of the symmetric property of
Cubic Structure of Si. But real devices are built near the surface hence the
orientation of the crystal does matter.
In 111 crystal terminates on 111 plane and in 100 it terminates on 100
plane. 111 plane has largest number of Si atoms per cm2 whereas 100 has
the least number of atoms per cm2. Because of this difference 111 planes
oxidize much faster because the oxidation rate is proportional to the Silicon
atoms available for reaction.
But because the atom surface density is the highest the dangling bond
surface density is also the highest in 111 hence Si/SiO2 has superior
electrical properties in terms of interface states in 100. Interface states give
rise to 1/f noise or flicker noise. Because of this superiority all MOS
devices use 100 crystal orientation. But historically BJT have used 111
because 111 crystal growth is easier to grow by Czochralski method. But as
we move to sub-micron and deep and ultra-deep sub-micron BJT, 100
crystal orientation seems to be the crystal orientation of choice for BJT also.
Z
(0,0,a)
<— [111]plane
Norinal direction to
(a,0,0) the [111] plane is called <111>
[100]plane
X-axis is the direction
of orientation of plane[100]
and is denoted=100>
VA Figure 10. Silicon Wafer Orientation
X Miller Index is used to denote the plane
Crystal Growth-Bulk and Epitaxial Film-Part 4
Part 4 of Crystal Growth describes the three technologies of Epitaxial Film
Growth- Chemical Vapour Phase Deposition, Liquid Phase Epitaxy and
Molecular Beam Epitaxy. First two are for Si Epitaxial Film growth and
MBE is for compound semiconductor epitaxial films.
Crystal Growth-Bulk and Epitaxial Film_Part 4
EPITAXIAL FILM GROWTH.
There are two major epitaxial film growth technologies:
Silicon Epitaxial Film Growth Technology (Low End Technology hence
economical) and Compound Semiconductor Epitaxial Film Growth
Technology (High End Technology hence expensive and suitable for niche
applications).
Silicon Epitaxial Film Growth Technology are further divided into:
Chemical Vapour Phase Deposition(C VPD) Technology and Liquid Phase
Epitaxy(LPE). LPE is relatively cheaper but very toxic.
Compound Semiconductor Epitaxial Film Growth is achieved by Molecular
Beam Epitaxy System which is inordinately expensive and in India in a few
places only we have MBE systems namely TIFR(Mumbai), Solid State
Physics Laboratory(Delhi), Central Electronics Engineering Research
Institute (Pilani), II1T(Madras) and CSIO(Chandigarh).
In Table 3 we make a comparative study of LPE, CVPD and MBE.
Molecular Beam Epitaxy is carried out under Ultra High Vacuum
Conditions. This means we are working at 107° Torr where 1 Torr is 1mm
of Hg. For achieving Ultra High Vacuum we have to work in three stages:
Rotary Pump is used for achieving 10~Torr. Silicon Oil Vapour Pump is
used to achieve 10°Torr and Ion Pump is used to achieve Ultra High
Vacuum of 10°°Torr. This ultra high vacuum requirement makes MBE
equipment inordinately expensive. This equipment is imperative for
Photonic Devices.
CHEMICAL VAPOUR PHASE DEPOSITION(CVPD) OF EPITAXIAL
FILMS or Chemical Vapour Phase Epitaxy(CVPE).
In Figure 11 , we describe CVPD system for obtaining Si Epitaxial Films. A
controlled chemical vapour of precise chemical composition at a precisely
controlled flow rate is passed over silicon substrates. Si Substrates act as the
seed crystals. They are kept at precisely controlled temperature of 1270°C
by RF induction heating hence they are placed on Graphite Susceptors. This
gives a precise control of Si molecules, in vapour phase of partial pressure
P, impinging upon Si Substrate:
(Impingement Rate) F =
r= P =| 3.5x10** P(Torr) j
v(2mmkT) /m(gm) T (Kelvin)
molecules
em?—sec
RF induction coils
SS a a
5) = 2" = 8S
SiCl4 + 2H2 «> Si) +4HCl Vafers
eta ts
Laer Ye Graphite Susceptor
© CO) & *
x pS Diborane B2H6
aoe
ae ee AsH3 Arsenie
V2 ;
<2 PH3 Phosphine
oe
—— 0
xX 3 SiCl4 (Silicon Tetrachlonde)
Figure 11. Chemical Vapour Phase Epitaxy for growing
epitaxial films of Silicon.
Table 3. Comparative study
of LPE, CVPD and MBE
Very high
quality films
Relatively
Inexpensive
Difficult to grow abrupt
heterostructure
Unable to grow
immiscible alloys
Unsitable for
Quantum Devices
Near Equilibrium Growth
for
Precise Alloy Properties
Extremely
High purity films
Not suited for
heterostructures
High quality
materials
Atomically abrupt
interfaces
Techniques can grow films
with stiochometric
coefficient far
from equilibrium
By this technique all
quantum structures
can be Down
Expensive technique
because of ultva-ligh
s |
vacuum re quirement
CPVD is suitable for Homo-epitaxy but not for hetero-epitaxy. The
byproduct of this reaction is HCl as seen from the chemical equation of
reaction:
At 1270°C SiCl,(Silicon Tetrachloride) + 2H» ~ Sil +4HCI. 1.
This byproduct HCl can attack the Silicon Substrate and cause chemical
etching. Hence Chemical Reaction as depicted Eq.(1) is reversible.
\
Film
Growth
Rate
(microm min)
Graph 1. Film growth rate vs y(Mole Fraction)
If y(mole fraction = SiCl, conc./Total conc) is greater than 0.23 as seen
from Graph 1 , growth rate is negative. This means chemical etching of
Silicon Wafer. The negative growth rate is used for in-situ cleaning of
Silicon Wafers. This is one of the ways of chemical etching while preparing
Silicon Wafers from Silicon Ingot.
If y < 0.23 we have positive growth rate and Si vapour deposition takes
place as an epitaxial film. The growth rate is maximum at y = 0.1 . The
maximum growth rate is 5um/minute.
Since CVPD is carried out at 1270°C, there is the problem of out-diffusion
of Arsenic from the substrate into the epitaxial layer. Arsenic is used as
buried layer to reduce the series collector resistance of Vertical NPN
transistor. Arsenic has very low diffusion coefficient still at the elevated
temperature of 1270°C, some out-diffusion is inevitable. To prevent this
out-diffusion altogether, we use Liquid Phase Epitaxy which is shown in
Figure 12.
LIQUID PHASE EPITAXY.
The Liquid Phase Epitaxy set up is shown in Figure 12.This is carried out at
900°C hence the problem of out-diffusion is completely prevented but
Silane which is used for LPE is highly toxic. As seen in Figure 12, there is a
super saturated solution of Silane kept at 900°C. By Graphite Slider, the
silicon substrate is brought directly underneath the well containing Silane.
As Substrate comes in contact with Silane , latter is dissociated into Silicon
and Hydrogen and Silicon precipitates onto the substrate forming an
epitaxial film on the substrate.
ha
Perspective View
Super Saturated Solution of Silane at
900 degree centigrade
Graphite Block
Si Substrate
Graphite Slider
Cross-sectional View
Graphite Block
Graphite Slider
Si Substrate
Super Saturated Solution of Silane at
900 degree centigrade
Figure 12. Liquid Phase Epitaxy Setup.
MOLECULAR BEAM EPITAXY
Due to explosive demand of Wireless Communication Equipment there has
been a sudden spurt in demand of MBE equipments particularly of multiple
wafer MBE equipment which can give a high throughput of epitaxilly
prepared wafers.
There are two kinds of epitaxial film growth: Homoepitaxy (same
composition) and Heteroepitaxy (different composition). By Molecular
Beam Epitaxy, multi-layered thin films of single crystals of different
compositions and of atomic dimension can be grown. In effect we can
achieve heteroepitaxy which is the hallmark of compound semiconductor
devices such as Photonic Devices and GaAs MESFET.
MBE is a process for making compound semiconductor materials with great
precision and purity. These materials are layered one on top of the other to
form semiconductor devices such as transistors and lasers. These devices
are used in such applications as fiber-optics, cellular phones, satellites,
radar systems, and display devices. MBE is used for fabrication of Super-
lattices and hetero-junction MESFET. Super lattices are periodic structures
of alternating Ultra-thin layers of compound semiconductor.
MBE growth produces complex structures of varying layers which are
further processed to produce a range of electronic and optoelectronic
devices, including high speed transistors, light-emitting diodes, and solid
State lasers. MBE is a powerful technique both for research into new
materials and layer structures, and for producing high-performance devices.
Effusion a As dopent
ra
i Sapphire View Port
Effusion Furnace
— SC 3 Liquid Nitrogen
: NL %
Main Shutter Cryogenic Panel
Flourscent Screen
To variable speed
motor & substrate
heater supply
tH |_| lal | Ionization Guage for
a ||| measuring pressure
upto 10°-11 Torr
View Port J
‘iew Port
Sample Exchange —— <Gate Valve
Load Lock
et TT
Figure 13. Molecular Beam Epitaxy Set Up.
Referring to Figure 13:
The walls of the chamber cooled with liquid nitrogen. This cryogenic
screening around the substrate minimizes the spurious flux of
contaminating atoms and
Sample Exchange Load Lock-this permits maintenance of Ultra High
Vacuum while changing the substrate.
Effusion Cells- this contains the solid source to be evaporated and deposited
on the substrate. Temperature of the effusion oven is adjusted to give
desired evaporation rate.
Rotating Substrate Holder- rotation of substrate ensures less than +1%
doping variation and +0.5% in thickness variation.
RHEED Gun-Reflection High Energy Electron Diffraction Gun gives a
beam of electron which can be made incident on the epitaxially grown film.
The diffraction pattern of electrons is studied on the fluorescent screen. This
shows a maximum when there is a completed monolayer and a minimum
when there is a partial layer as this produces more scattering. Thus RHEED
Gun is used for in-situ monitoring of the growth of the epitaxial film mono-
layer by mono-layer.
Computer controlled shutters of each furnace allows precise control of the
thickness of each layer, down to a single layer of atoms.
Substrate Heating- to obtain high quality epitaxial layer, growth
temperature must be relatively high. The substrate wafer must be heated to
allow the atoms to move about the surface and reach the proper ordered
site. Growth must be at a temperature where growth rate is insensitive of
minor temperature variation. Atoms arriving at the substrate surface may
undergo absorption to the surface, surface migration, incorporation into the
crystal lattice, and thermal desorption. Which of the competing pathways
dominates the growth will depend strongly on the temperature of the
substrate. At a low temperature, atoms will stick where they land without
arranging properly - leading to poor crystal quality. At a high temperature,
atoms will desorb (reevaporate) from the surface too readily - leading to
low growth rates and poor crystal quality. In the appropriate intermediate
temperature range, the atoms will have sufficient energy to move to the
proper position on the surface and be incorporated into the growing crystal.
The Inventors of MBE are : J.R. Arthur and Alfred Y. Chuo (Bell Labs,
1960).
MBE is a technique for epitaxial growth of single crystal atomic layer films
on single crystal substrate. It gives precise control in chemical composition
and doping profiles. To avoid contamination of the epitaxial films, Ultra
High Vacuum within MBE chamber is imperative. It requires Very/Ultra
high vacuum (10° Pa or 10°!'Torr). This implies that Epitaxial film is
grown at slow deposition rate (1 micron/hour). This permits epitaxial
growth of single atomic layer if desired. Slow deposition rates require
proportionally better vacuum. Ultra-pure elements are heated in separate
quasi-knudson effusion cells (e.g., Ga and As) until they begin to slowly
sublimate. Gaseous elements then condense on the wafer, where they may
react with each other (e.g., GaAs). The term “beam” in MBE means the
evaporated atoms do not interact with each other or with other vacuum
chamber gases until they reach the wafer. Each gas beam may be turned on
and off rapidly with a shutter or a valve. Beam intensity (called the flux) is
adjusted for precise control of layer composition. A collection of gas
molecules moving in the same direction constitute the molecular beam.
Simplest way to generate a molecular beam is Effusion cell or Knudsen cell
. Oven contains the material to make the beam. Oven is connected to a
vacuum system through a hole. The substrate is located with a line-of-sight
to the oven aperture. From kinetic theory, the flow through the aperture is
simply the molecular impingement rate on the area of the orifice.
Impingement rate is: The total flux through the hole. The spatial
distribution of molecules from the orifice of a knudsen cell is normally a
cosine distribution. The intensity drops off as the square of the distance
from the orifice. Intensity is maximum in the direction normal to the orifice
and decreases with increasing 8, which causes problems. Use collimator, a
barrier with a small hole; it intercepts all of the flow except for that
traveling towards the sample.
During the MBE process, growth can be monitored in situ by a number of
methods:
Reflection high energy electron diffraction (RHEED), using forward
scattering at grazing angle, which shows a maximum when there is a
completed monolayer and a minimum when there is a partial layer as this
produces more scattering;
Low energy electron diffraction (LEED), takes place in backscattering
geometry and can be used to study surface morphology, but not during
growth;
Auger electron spectroscopy (AES), records the type of atoms present;
Modulated beam mass spectrometry (MBMS), allows the chemical species
and reaction kienetics to be studied.
Computer controlled shutters of each furnace allows precise control of the
thickness of each layer, down to a single layer of atoms.
Intricate structures of layers of different materials can be fabricated this
way e.g., semiconductor lasers, LEDs.
Before starting the epitaxial growth, in-situ cleaning of substrate is
required. This is achieved by High Temperature Baking of the substrate.
This decomposes and vaporizes the oxide layer over the substrate. The
second method of in-situ cleaning is low energy ion beam of inert gas is
used to sputter clean the surface. After the sputtering , low temperature
anneal;ing is required to reorder the surface lattice system.
If there is a lattice mismatch between the substrate and the growing film,
elastic energy is accumulated in the growing film. At some critical film
thickness, the film may break/crack to lower the free energy of the film.
The critical film thickness depends on the Young’s moduli, mismatch size,
and surface tensions. Hence under heteroepitaxy, we must keep the
thickness lower than critical film thickness.
Figure 14 shows the physics of epitaxial growth in MBE system. . The aim
of this process is to enable sharp interfaces to be formed between one type
of alloy and the next e.g. GaAs and AlAs, and thus create structures which
may confine electrons and exibit 2-dimensional behaviour. Molecular Beam
Epitaxy (MBE) is basically a sophisticated form of vacuum evaporation.
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Figure 14. Molecular Beam Epitaxy growth mono-layer by
layer or atom layer by layer.
METAL ORGANIC CHEMICAL VAPOUR PHASE DEPOSITION>
The growth process in MOCVD (metal-organic CVD, also known as
MOVPE metal-organic vapour phase epitaxy) is similar to MBE, but the
atoms are carried in gaseous form to the substrate. GaAlAs growth is
achieved by using a mixture of hydrogen as a carrier gas and organometallic
precursors such as trimethyl galium and/or trimethyl aluminium together
with arsine. The growth rate can be 10 times greater than in MBE, the
process does not require ultra high vacuum and it can be scaled up from
research to production of commercial devices relatively easily. However,
the preparation of the gaseous mixtures has to be very carefully controlled
so that as yet it is unclear which technique will eventully dominate. One
advantage MOVPE has over MBE is in the ability to grow phosphorous
containing alloys, once phosphorous has been introduced into an MBE
chamber it is almost impossible to grom anything else! One disadvantage is
that in situ monitoring is more difficult.
Tutorial on Chapter 1-Crystal and Crystal Growth.
Tutorial on Chapter 1 gives the typical numerical problems in Crystal
Growth-Bulk and Epitaxial.
Tutorial on Chapter 1-Crystal and Crystal Growth.
1. Give the percentage packing of Face-Centered-Cube ? [Ans.74%]
Method:
FCC crystal has 8 atoms at the 8 corners of the cube and 6 atoms on the 6
faces (4 on side faces+2 on the upper and lower faces].
Therefore N = the coordination Number = the number of atoms per unit cell
= (8x1/8+6x1/2)= 4 atoms per unit cell.
The atoms are solid spheres and are closely packed so that the atom at the
center of the face are touching the 4 atoms on the four corners of the face as
shown in the Figure I.
‘a' Angstrom
WO.1jssuy PB,
Arrangement of the corner spheres and face centered sphere.
Inspecting the Figure we find that:
4R=diagonal of the face=av2.
Therefore R/a = V2/4=1/(2V2)
packing ratio = Sanne = ~. x 4 ERS = <n x ©: = ial ems 3= ame = = 0.74
Therefore packing percentage in FCC crystals is 74%.
1. Give the percentage packing of Body-Centered-Cube ? [Ans.68%]
Method:
BCC crystal has 8 atoms at the 8 comers of the cube and 1 atom at the
center of the Cube. Therefore N = the coordination Number = the number
of atoms per unit cell = (8x1/8+1)= 2 atoms per unit cell.
The atoms are solid spheres and are closely packed so that the atom at the
center of the Cube is touching the 2 atoms on the two corners of the cross
diagonal of the cube as shown in the Figure 2.
d*= cross diagonal of the CUBE
WO1jssuy FP,
‘a/2'Angstrom
Cross Diagonal of the lower face
Inspecting the Figure we find that:
4R= cross diagonal of the Cube =av3.
Therefore R/a = V3/4;
8 v3, 8 3V3 ¥3
” a 2V phere
packing ratio = a = 5 x5nr 8a onx()?=on a7) = 4¥xX = oe ost
Therefore packing percentage in FCC crystals is 68%.
1. Determine the conducting electrons density in FCC Cu where the
lattice parameter is 3.6 Angstrom ? [Ans. 8.5734x1078/m?]
Copper is univalent crystal hence each atom contributes one electron.
Therefore atomic density gives the conducting electron density.
Coordination Number of Cu = N = 4 atoms per unit cell.
Unit cell volume is a° .
Therefore atomic density = conducting electron density = 4 / (a?) =
8.5734x108/m?3 = 8.5734x107/cc.
1. Si, Ge and GaAs are all diamond crystal structure. Diamond structure
is two interpenetrating FCC crystals with one sub-lattice displaced
w.r.t. the other along the cross diagonal of the cube by a quarter of the
cross diagonal length. In case of Ga As it is ZincBlende cubic structure
which has one sub-lattice of Ga and the other sub-lattice of As.
Find the weight density of Si,Ge and GaAs. Given Avogadro Number =
Nayo =6.02*10*%atoms/gm-mole, Si, Ge and GaAs have atomic weight of
28.1, 72.6 and 144.63(mean=72.315 respectively and the lattice parameters
are 5.43A°, 5.646A° and 5.6533A° respectively.
[Ans. 2.33gm/cc, 5.37gm/cc and 5.32gm/cc respectively]
Method:
Coordination Number =N= 8 corner atoms + 6 face center atoms + 4 body
center atoms= (1/8x8 + 1/2x6+ 4)= 8 atoms per unit cell.
Therefore Number density = N* = 8/a°= 4.9967x10*? atoms/cc, 4.445 x10?
atoms/cc, 4.4277 x1022 atoms/cc,of Si, Ge and GaAs respectively.
Weight of one atom= (AW gm/mole)+ (Nay, atoms/mole)= 4.6610°79
gm/atom. 1.157x10°** gm/atom. 1.244x10°°* gm/atom of Si, Ge, GaAs
respectively.
Therefore weight density = Wt of one atomx N” = Weight Density = p
Therefore density of Si=4.9967x10*? atoms/ccx4.6677* 10°79
gm/atom=2.33gm/cc.
Therefore density of Ge=4.4495x10*? atoms/ccx1.206* 10°22
gm/atom=5.366gm/cc.
Therefore density of GaAs=4.42776x10** atoms/ccx1.2012* 10°72
gm/atom=5.318gm/cc gm/cc.
1. A Silicon Ingot should have a P-Type doping of 10!°Boron Atoms/cc.
In a Czocharlaski Crystal Growth set-up what amount of Boron
element by weight should be added to a Si melt of 60Kg.
Given:Atomic Weight of Boron =10.8, Segregation Coefficient= 0.8.
[Answer; About 6 milli gm]
Method:
Let Solid Phase concentration of Boron = Cz ;
Let Liquid Phase concentration of Boron = Cy, ;
By Definition, segregation coefficient = Cs / Cy, = 0.8.
Therefore C;, = (10!°/cc)/0.8= 1.25x10!°atoms/cc.
Volume of the Melt = Vo = (Weight of the Si-Melt)/ps; = (60*10°
gm)/(2.33gm/cc).
Therefore Vo = 25.75*10° cc.
No. of Boron atoms to be added= Vox Cy, = 25.75*10° cc
x 1.25x10!®atoms/cc=Np.
Therefore No = 32.2x10!9 Boronatoms to be added to the whole melt.
Number of gm-moles of Boron required = No /Nayogadro
= 32.2x10!9 Boron Atoms/6.021073 Boron Atoms/gm-mole = 5.349104
gm-mole.
Weight of Boron required=
gm-molexAtomic Weight =5.349x10-* gm-molex10.8gm/gm-mole= 5.78
milligm.
1. A layer of 10pm thick epitaxial film is to be deposited on a Si
Substrate by CVD method. For how long should the substrate stay in
the CVD oven at 1270°C if the gas mixture of gases is in the following
concentrations in atoms/cc: H> is 2.94109 and SiCl, is 6x10".
Given:
Deposition temperature, 1270° C
H, flow, one liter/min
~W
n eC & OF @
Silicon deposition in microns/min
a
De
:
a:
fe
7
A
a
i
‘Al
TT
if
Mol fraction SiCI, in H,
The above figure gives Growth Rate(microns/min) vs the Mole Fraction(y)
of SiCl,. At higher concentrations of HCl i.e. at mole fraction > 0.275 in-
situ etching occurs.
From the Figure the following Table is generated:
Mole Fraction Growth Rate(microns/min)
0.025 1.6
0.050 3.8
0.075 4.6
0.100 5.0
0.125 4.5
0.150 3.6
0.175 Paee
0.200 2.1
0.225 1.3
0.250 0.6
0.275 0
0.300 -0.8
In the above table the growth rate is the net result of two competing
reactions:
Forward reaction where SiCl, is decomposing at 1270°C and aiding the Si
epitaxial growth and Backward reaction where HCl vapour is etching the Si
surface in-situ.
From low mole fraction to 0.275, forward reaction dominates. At 0.275 the
two competing reactions are balanced. Above mole fraction 0.275 backward
reaction prevails resulting into etching of the Si surface of the substrate.
In our problem the Mole Fraction is :
Conc. of Sicl,
ee
Conc. of H,+Conc.of SiCcl,
Mole Fraction =
From the Graph, corresponding to y= 0.02, the growth rate is 1.5microns
per minute.
Therefore 10 microns will require 6.7 minutes of exposure in CVD Furnace.
1. In a Molecular Beam Epitaxy System, find the time required to form a
monolayer of O» at pressure 1 Torr, 10°°Torr and 10°!°Torr. Given
Oxygen Molecular Weight = M=32 and Oxygen Molecular Diameter =
3.45A°. Assume that the mono-layer is a close pack structure.
[Answer: 2.43x10° sec, 2.43 sec, 6 hours]
Method:
Surface Concentration of the closely packed monolayer of Oxygen
Molecules= Ns .
Let the time required for monolayer of Oxygen formation be ty where
to=Ns/®
Where ®=Impingement Rate of Oxygen molecules on Substrate Surface in
MBE Bell Jar.
P 3.51 10” x P*(Torr)
2nMkT JM X T(Kelvin)
g=
molecules /(cm*.sec)
Here M=32 and T =300K and P is the corresponding pressure in Torr at
which the time for monolayer growth is to be calculated.
P(Torr) (impingement rate) ty (units of time)
1 3.58x 107° 2.8
1x10°6 3.58x10!4 2.8sec
1x10°10 3.58x1010 27914s=7.75hours
This problem clarifies why MBE requires a super Vacuum to fabricate
stable and reproducible devices. This is precisely why using Compound
Semiconductors is prohibitively costly and we are looking for alternatives
to Compound Semiconductors.
Syllabus of EC1419A_Electrical and Electronic Materials Science
EC1419A is a 3 credit course in Materials Science with 42 periods per
Semester.
EC 1x19A Electrical & Electronic Material
Revised Syllabus with effect from session 2011-12
L-T-P: 3-0-0 Credit: 3
1. Band Theory of Solids- Energy Band Diagram, E-k diagram, reduced E-k
diagram, insulators, semiconductors and conductors [6 lectures]
2. Dielectric Behaviour of Materials: polarization, dielectric constant at low
frequency and high frequency, dielectric loss, piezo electricity and ferro
electricity [6 lectures];
3. Magnetic Behaviour of Materials- diamagnetism, para magnetism,
ferromagnetism and ferrimagnetism; soft and hard magnetic materials and
their applications [6 lectures]
4. Semiconductor- single crystal, polycrystal and amorphous; Fermi-Dirac
Distribution; Hall Effect; Intrinsic and Extrinsic; N-type and P-type; Crystal
Growth-(1) preparation of electronic grade polycrystal in Siemen‘s Reactor
(2) Czochrarloski Method and Float Zone Method of bulk single crystal
ingot preparation(3) mirror finished wafer preparation (4) Epitaxial Film
Growth- Chemical Vapour Phase Deposition & Liquid Phase Epitaxy (5)
Molecular Beam Epitaxy [12 lectures]
5. Concept of Phonons- quantization of lattice vibration [2 lectures]
6. Special classification of Semiconductor Materials- degenerate (semi-
metal) and non-degenerate semiconductor; elemental and compound
semiconductor; direct and indirect band gap material [3 lectures]
7. Superconductors- low and high temperature (YBaCuO) superconductors;
Meissner Effect, applications [3 lectures]
8. Special Materials: Nano materials (ZnO, TiO2, buckeyball carbon and
graphene), semiconducting polymers, flexible electronic materials, meta
materials, smart materials.
Text Book:
References:
1. Principles of Electric Engineering Materials and Devices by S.O. Kasp,
McGraw Hill;
2. Structures and Properties of Materials Vol VI, electronic properties by
Robert M. Rose, Lawrence A. Shepherd and John Wulf, Wiley Eastern Itd;
Chapter 2. Solid State of Matter.
Chapter 2 describes the Solid State of Matter, its four classes namely Metals, Ceramics,Polymers and Composites.
The electronic configuration of Inert Gases are described and their applications pointed out.
Chapter 2. Solid State of Matter.
"It is evident that many years of research by a great many people,
both before and after the disacovery of the transisitor effect has been required
to bring the knowledge of semiconductors to the present development.
We were fortunate to be involved at a particulary opportune time and to add another
small step in the control of Nature for the benefit of mankind.
- John Bardeen, 1956 Nobel Lecture.
We were taught in our high school that there are three states of matter:
Solid- it has a fixed volume and fixed shape.
Liquid- it has fixed volume but it takes the shape of the vessel it is kept in.
Gas-it has no fixed volume and no fixed shape. It takes the volume and shape of the vessel it is kept in.
From this we concluded that in Gaseous State inter-molecular distance is not fixed and the arrangement of the
molecules is not fixed. Whereas in Liquid State inter-molecular distance is fixed but the arrangement of the
molecules is not fixed. In contrast in Solid State inter-molecular distance is fixed and as well as the arrangement of
the molecules is fixed. In fact every elemental or compound solid has a well defined crystalline structure. Every
Solid has a characteristic Unit Cell and this is periodically repeated at an spatial distance known as Lattice
Parameter.
The most commonly found Unit Cell structures are: Cubic Cell, FCC Cell and BCC Cell.
A large mm range spatial periodicity is called Single Crystal.
Micron range spatial periodicity is known as Poly-crystal.
Nanometer range spatial periodicity is known as Amorphous.
In Figure 2.1. single crystal, poly crystal and amorphous solids are shown .
ee ee
=S—= (wea
(a) Single Crystal of mm dimensions (b) Poly-crystal with micron size grains (c)mass of minute crystallites
amorphous
Figure 2.1. Comparative Study of Single crystal, Poly- crystal and Amorphous Solid State of Matter
We subsequently learnt that there was a FOURTH STATE of Matter namely PLASMA. Matter above 4000K is in
plasma state. The whole Universe was in a state of plasma up to 380,000 years after the Big-Bang.
Now we know a FIFTH STATE of matter called Bose-Einstein Condensate at a fraction of Kelvin.
Section 2.1. Alloy- Solid Solutions.
Only a few elements are widely used commercially in their pure form. Generally, other elements are present to
produce greater strength, to improve corrosion resistance, or simply as impurities left over from the refining
process. The addition of other elements into a metal is called alloying and the resulting metal is called an alloy.
Even if the added elements are nonmetals, alloys may still have metallic properties.
Copper alloys were produced very early in our history. Bronze, an alloy of copper and tin, was the first alloy
known. It was easy to produce by simply adding tin to molten copper. Tools and weapons made of this alloy were
stronger than pure copper ones. The typical alloying elements in some common metals are presented in the table
below.
Table 2.1. Some Important Alloys and their constituent elements.
Alloys Composition
Brass Copper,Zinc
Bronze Copper, Zinc, Tin
Pewter Tin, Copper, Bismuth, Antimony
Cast Iron Iron, Carbon, Manganese, Silicon
Steel Iron, Carbon(plus other elements)
Stailess Steel Iron, Chromium, Nickel
The properties of alloys can be manipulated by varying composition. For example steel formed from iron and
carbon can vary substantially in hardness depending on the amount of carbon added and the way in which it was
processed.
Section 2.2. General Material Classification.
Based on atomic bonding forces, matter is classified in three classes namely: metallic, ceramic and polymeric.
These three classes can be further combined together to form composites. In Figure 2.2. we illustrate the three
classes and their composites.
Metals: : Polymeric:
(1)Ferrous Metals and Alloys. (1)Thermoplastics,
(2)Non-ferrous Metals and (2)Thermoset Plastics,
Alloys- (3)Elastometers.
Nickel, Titanium,precious
metals, refractory Composite:
metals,superalloys. (1)Re-inforced Plastics,
Refractory Metals are highly (2)Metal-matrix composites,
resistant to temperature for (3)Ceramic-matrix composites,
example:Niobium,Molybdenum, (4)Sandwich Structure,
Tantulum,Tungsten,Rhenium. (4)Concrete.
Ceramics:
(1)Glasses,
(2)Glass Ceramic,
(3)Graphite,
(4)Diamond.
Figure 2.2.The three classes of matter and their composites.
Section 2.2.1. Metals.
Metals account for about two thirds of all the elements and about 24% of the mass of the planet. Metals have
useful properties including strength, ductility, high melting points, thermal and electrical conductivity, and
toughness. From the periodic table, it can be seen that a large number of the elements are classified as being a
metal. A few of the common metals and their typical uses are presented below.
Common Metallic Materials:
(1)Iron/Steel - Steel alloys are used for strength critical applications
(2)Aluminum - Aluminum and its alloys are used because they are easy to form, readily available, inexpensive,
and recyclable.
(3)Copper - Copper and copper alloys have a number of properties that make them useful, including high electrical
and thermal conductivity, high ductility, and good corrosion resistance.
(4)Titanium - Titanium alloys are used for strength in higher temperature (~1000° F) application, when component
weight is a concern, or when good corrosion resistance is required
(5)Nickel - Nickel alloys are used for still higher temperatures (~1500-2000° F) applications or when good
corrosion resistance is required.
(6)Refractory materials are used for the highest temperature (> 2000° F) applications.
In metal, the lattice centers are immersed in a sea of conducting electrons. These conducting electrons give
metallic bonding which is malleable and ductile. It does not have brittleness. Plus it is conductive.
Section 2.2.2. Ceramics.
A ceramic has traditionally been defined as “an inorganic, nonmetallic solid that is prepared from powdered
materials, is fabricated into products through the application of heat, and displays such characteristic properties as
hardness, strength, low electrical conductivity, and brittleness." The word ceramic comes the from Greek word
"keramikos", which means "pottery." They are typically crystalline in nature and are compounds formed between
metallic and nonmetallic elements such as aluminum and oxygen (alumina-Al,O03), calcium and oxygen (calcia -
CaO), and silicon and nitrogen (silicon nitride-Si3N,).
The two most common chemical bonds for ceramic materials are covalent and ionic. Covalent and ionic bonds are
much stronger than in metallic bonds and, generally speaking, this is why ceramics are brittle and metals are
ductile.
Section 2.2.3. Polymers.
A polymeric solid can be thought of as a material that contains many chemically bonded parts or units which
themselves are bonded together to form a solid. The word polymer literally means "many parts." Two industrially
important polymeric materials are plastics and elastomers. Plastics are a large and varied group of synthetic
materials which are processed by forming or molding into shape. Just as there are many types of metals such as
aluminum and copper, there are many types of plastics, such as polyethylene and nylon. Elastomers or rubbers can
be elastically deformed a large amount when a force is applied to them and can return to their original shape (or
almost) when the force is released.
Polymers have many properties that make them attractive to use in certain conditions. Many polymers:
(1)are less dense than metals or ceramics,
(2)resist atmospheric and other forms of corrosion,
(3)offer good compatibility with human tissue, or
(4)exhibit excellent resistance to the conduction of electrical current.
The polymer plastics can be divided into two classes, thermoplastics and thermosetting plastics, depending on how
they are structurally and chemically bonded. Thermoplastic polymers comprise the four most important
commodity materials — polyethylene, polypropylene, polystyrene and polyvinyl chloride. There are also a number
of specialized engineering polymers. The term ‘thermoplastic’ indicates that these materials melt on heating and
may be processed by a variety of molding and extrusion techniques. Alternately, ‘thermosetting’ polymers can not
be melted or remelted. Thermosetting polymers include alkyds, amino and phenolic resins, epoxies, polyurethanes,
and unsaturated polyesters.
Rubber is a natural occurring polymer. However, most polymers are created by engineering the combination of
hydrogen and carbon atoms and the arrangement of the chains they form. The polymer molecule is a long chain of
covalent-bonded atoms and secondary bonds then hold groups of polymer chains together to form the polymeric
material. Polymers are primarily produced from petroleum or natural gas raw products but the use of organic
substances is growing. The super-material known as Kevlar is a man-made polymer. Kevlar is used in bullet-proof
vests, strong/lightweight frames, and underwater cables that are 20 times stronger than steel.
Section 2.2.4. Composites.
A composite is commonly defined as a combination of two or more distinct materials, each of which retains its
own distinctive properties, to create a new material with properties that cannot be achieved by any of the
components acting alone. Using this definition, it can be determined that a wide range of engineering materials fall
into this category. For example, concrete is a composite because it is a mixture of Portland cement and aggregate.
Fiberglass sheet is a composite since it is made of glass fibers imbedded in a polymer.
Composite materials are said to have two phases. The reinforcing phase is the fibers, sheets, or particles that are
embedded in the matrix phase. The reinforcing material and the matrix material can be metal, ceramic, or polymer.
Typically, reinforcing materials are strong with low densities while the matrix is usually a ductile, or tough,
material.
Some of the common classifications of composites are:
e Reinforced plastics
e Metal-matrix composites
e Ceramic-matrix composites
e Sandwich structures
e Concrete
Composite materials can take many forms but they can be separated into three categories based on the
strengthening mechanism. These categories are dispersion strengthened, particle reinforced and fiber reinforced.
Dispersion strengthened composites have a fine distribution of secondary particles in the matrix of the material.
These particles impede the mechanisms that allow a material to deform. (These mechanisms include dislocation
movement and slip, which will be discussed later). Many metal-matrix composites would fall into the dispersion
strengthened composite category. Particle reinforced composites have a large volume fraction of particle dispersed
in the matrix and the load is shared by the particles and the matrix. Most commercial ceramics and many filled
polymers are particle-reinforced composites. In fiber-reinforced composites, the fiber is the primary load-bearing
component. Fiberglass and carbon fiber composites are examples of fiber-reinforced composites.
If the composite is designed and fabricated correctly, it combines the strength of the reinforcement with the
toughness of the matrix to achieve a combination of desirable properties not available in any single conventional
material. Some composites also offer the advantage of being tailorable so that properties, such as strength and
stiffness, can easily be changed by changing amount or orientation of the reinforcement material. The downside is
that such composites are often more expensive than conventional materials.
Section 2.3. The atomic bonding in Matter.
It should be clear that all matter is made of atoms. From the periodic table, it can be seen that there are only about
100 different kinds of atoms in the entire Universe. These same 100 atoms form thousands of different substances
ranging from the air we breathe to the metal used to support tall buildings. Metals behave differently than
ceramics, and ceramics behave differently than polymers. The properties of matter depend on which atoms are
used and how they are bonded together.
There are 4 kinds of atomic bonding:
i.Metallic Bonding.
ii-Covalent Bonding.
iii.lonic Bonding.
iv. Van-der-Waal Bonding.
All chemical bonds involve electrons. Atoms will stay close together if they have a shared interest in one or more
electrons. Atoms are at their most stable and chemically inert form when they have no partially-filled electron
shells such as Inert Gases /Noble Gases/Rare Gases such as He,Ne, Ar, Kr, Xe and Rn . These are odourless,
colorless, monoatomic gases with very little reactivity hence they are called Inert Gases.
In Table 2.2 and Table 2.3. the salient parameters and the electronic shell configurations of the Noble Gases are
given.
Table 2.2. The salient parameters of Noble Gases
Atomic
Gas B.P(K) M.P.(K) Z* radius(pm) IonizationEnergy(eV)
Gas
B.P(K)
4.4Below 4.4K He
exhibitssuperluidity
27.3
87.4
121.5
166.6
211.5
*Z, = Atomic Number.
M.P.(K)
0.95Solid He found
in the Core of
Jupiter.It Behaves
like metal
24.7
Table 2.3. Shell Structure of Inert Gas Atoms.
K-
Z Shell(n=1)
2 1s2
10 1s2
18 1s2
36 1s2
54 1s2
86 1s2
tT
Shell(n=2)
2s? ,2p®
2s* ,2p®
2s* ,2p®
2s? ,2p®
287 ,2p°
M-
Shell(n=3)
3s? ,3p®
382
Bp ad
382
ap ad”
382
apo ad-”
Table 2.4. Simplified Shell Structure of Inert Gas Atoms.
Gas
1s2
Atomic
radius(pm)
31
38
71
88
108
120
N-Shell(n=4)
4s? ,Ap®
4s? ,Ap®,4d'
4s?
Ap*.4d'?,4f"*
IonizationEnergy(eV)
24.8
O-
Shell(n=5)
5s? ,5p®
582
sep od
P-
Shell(n=6)
6s? ,6p®
Gas Z
Ne 10 He De? 2p?
Ar 18 Ne 33° 3p°
Kr 36 Ar 3d’ 4s? ,Ap®
Xe 54 Kr 4d10 5s? ,5p®
Rn 86 Xe 4fl4 5d!0 6s? ,6p®
K, L, M, N, O and P-Shell are major Shells corresponding to the Principal Quantum Number n = 1,2,3,4,5 and 6.
Within each Shell there are Sub-Shells s, p, d and f.
Here we will briefly discuss the Periodic Table.
Principal Quantum Number: n — gives the energy quantization as well as the complete ONE period of elements.
Azimuthial Quantum Number: |— gives the orbital angular momentum quantization.
Magnetic Quantum Number: m = JA, (I-1)h...... ere -(I-1)h, -lh. This gives the orientation quantization. When
a magnetic field Bexternal is applied in Z-axis, Orbital Angular Momentum L will align so as to give an Integral
Projection of [h on Z-axis as shown in Figure 2.2.
1= 2, |LJ= nv /+1)]
Z-axis ———= axis of external
eee magnetic field
Figure 2.2. The Projection of the orbital angular momentum
|L| along the Z-axis along which an external Magnetic Field is
applied.
Here for K-Shell, corresponding to Principal Q.N. n=1, | can be only 0.There is only s-subshell.
For L- Shell, corresponding to Principal Q.N. n=2, | can be 0 and 1.Here there are two subshells namely: s-
subshell and p-subshell.
For M- Shell, corresponding to Principal Q.N. n=3, | can be 0 ,land 2. Here there are three subshells namely: s-
subshell , p-subshell and d-subshell.
For N- Shell, corresponding to Principal Q.N. n=4, | can be 0 ,1, 2 and 3. Here there are four subshells namely: s-
subshell , p-subshell , d-subshell and f-subshell.
For O- Shell, corresponding to Principal Q.N. n=5, | can be 0 ,1, 2 , 3 and 4. Here there are five subshells namely:
s-subshell , p-subshell , d-subshell , f-subshell and g-subshell.
For P- Shell, corresponding to Principal Q.N. n=6, | can be 0 ,1, 2,3, 4.and 5. Here there are six subshells
namely: s-subshell , p-subshell , d-subshell , f-subshell , g-subshell and h-subshell.
In nut-shell,the above information can be summarized by Table 2.5.
Table 2.5. Electron Distribution among the Sub-Shells.
Sub-ShellOr Permissible
Orbital | ml statesw/o spin Orbitalshape
Ss 0 0 1 SPHERE
TWODUMB
P ei) See : BELLS
FOURDUMB
d 2 -2,-1,0,+1,+2 5 BELLS
; 3 B21, : EIGHTDUMB
0,+1,+2,+3 BELLS
Name
sharp
principal
diffuse
fundamental
The ORBITAL SHAPES of electron cloud for s-orbital, p-orbital, d-orbital and f-orbital are illustrated in Figure
2.3%
n=1 l=0 *% s orbital
|e eee
P tet | D@® | porita |
ee
| tet | G@® [| pordita |
| i |
d orbital
ee [| sorbat —
oe x ee
ee
[13 [complex stape | — fort —
Figure 2.3. Electronic Configuration or the shape of
the Electron Cloud surrounding the Nucleus for
different values of azumuthial quantum number.
Spin Quantum Number: s = +(1/2)h. This gives the quantization of the spin angular momentum. This Quantum
Number ‘s’ has been illustrated in Figure 2.4.
S. (along B.) Spin Up
— I,
+h/2 m,=+"/
-h/2
Spin Down
Figure 2.4. Electron, being a FERMION, has an axial spin angular
momentum (/3/2)h which has a projection of +(h/2) on Z-axis.
Z-axis is the axis of externaly applied Magnetic Field.
For each unique set of (n, L, m) there can be two permissible electronic states:one corresponding to +(1/2)h and
the second anti-parallel —(1/2)h.
Pauli Exclusion Principle clearly states that no two electrons can have the same four quantum numbers. Atleast
one quantum number should differ.
For n = 1,1 can only be 0. This marks the First Period. This means spherically symmetrical electron cloud
surrounding the nucleus. This means Orbital Angular Momentum will be always ZERO in first period.
Therefore in First Period, n=1, | = 0, m = 0, s = + (1/2)h — this correspond to TWO elements H and He.
H is the first Group and He is the last Group in First Period. This is illustrated in Table 2.6.
Table 2.6. Elements of FIRST Period.
pnd grd qth 5th gth 7th
FirstPeriod 15‘ Gr Gr Gr Gr Gr Gr Gr 8 Gr
Elements H - - - - - - He
s-orbitalbeing i 7 7 - ° - 252
filled
FirstPeriod 1°* Gr Gr Gr Gr
K-Shellls being s-subshellin
Filled up K-Shell
For n=2, | can be 0 and 1. This marks the Second Period.
5th gth
7th
Gr gh Gr
s-subshellin K-
Shellis Full
Here we have K Shell which is already FULL and corresponds to He configuration and L-Shell is in the process of
getting filled as shown in Table 2.7.
Table 2.7. Elements of SECOND Period.
SecondPeriod 15 Gr 2nd Gr 3rd Gr
Elements Li Be B
He
corresponds to He2s! He2s? He2s*2p!
K-Shell
S-
L-Shell Is S- , p-
being filled subshellin ae subshellin
up. L-Shell Fal ae L-Shell
For n=3, | can be 0, land 2. This marks the Third Period.
4h Gr
He2s?2p*
subshellin
L-Shell
5th Gr
N
He2s?2p3
p-
subshellin
L-Shell
6h Gr
He2s*2p*
p-
subshellin
L-Shell
Here we have K Shell which is already FULL and L-Shell is also filled and this K and L filled Shell corresponds to
Neon. M-Shell is in the process of getting filled up as shown in Table 2.8.
Table 2.8. Elements of THIRD Period.
ThirdPeriod 15‘ Gr 2nd Gr 3rd Gr
Elements Na Mg Al
Ne
corresponds
to K- Ne3s! Ne3s? Ne3s?3p!
ShellAnd L-
Shell
Ne3s*3p?
Ne3s*3p*
Ne3s*3p*
ThirdPeriod 15* Gr 2nd Gr 3° Gr 4'h Gr 5 Gr 6" Gr
xs
M-Shell Is s- a helli p- p- p- p- E
being filled subshellin a Soe subshellin subshellin subshellin —subshellin —s
up. M-Shell a en'S M-Shell M-Shell M-Shell M-Shell D
For n=4, | can be 0, 1, 2 and 3 This marks the Fourth Period.
Here we have K Shell and L-Shell are filled up. But M-Shell is partially filled up. K,L, and M(partially)
corresponds to Ar. Partially filed M-Shell and empty N-Shell is in the process of getting filled up as shown in
Table 2.9.
Table 2.9. Elements of FOURTH Period.
FourthPeriod 15t Gr 2nd Gr + 34 Gr 4h Gr 5th Gr 6h Gr
Elements K Ca Ga Ge As Se
Ar
corresponds
to K, L-Shell 10402 1042 104c2 104,
filled upAnd Ar4s! Ar4s* ei as pe cote ‘a Cos ie oes
M-Shell P P P P
partially filled
up.
N-Shell, s-
orbital is first d- d- d- d-
filled S- S subshell subshell subshell subshell
up.TThen M- subshellin subehellan in M- in M- in M- in M-
Shell d-orbital N-Shell : Shell is Shell is Shell is Shell is
er 5 N-Shellis
is filled getting Full fullp- full p- fullp- fullp-
up.Then N- filled subshellin subshellin subshellin subshelli
Shell p-orbital N-Shell N-Shell N-Shell N-Shell
is filled up
T Sc, Ti, V,Cr,Mn,Fe,Co,Ni,Cu,Zn are the d-Block Transition elements which are there due to belated filling up of
d-orbitals in M-Shell which has 10 electron states permissible from 3d! to 3d! .
For n=5, |! can be 0, 1, 2, 3 and 4. This marks the Fifth Period.
Here we have K Shell , L-Shell and M-Shell are filled up. But N-Shell is partially filled up. K,L, M and
N(partially) corresponds to Kr. Partially filed N-Shell and empty O-Shell is in the process of getting filled up as
shown in Table 2.10.
Table 2.10. Elements of FIFTH Period.
FifthPeriod
Elements
Kr
corresponds
to K, L,M-
Shell filled
upAnd N-
Shell
partially
filled up.
O-Shell s-
orbital Is
first filled.T
Then N-
Shell is
filled up
and then O-
Shell is
being filled
up.
1°t Gr
Rb
Kr5s!
S-
subshellin
O-Shell
getting
filled
2nd Gr
Sr
Kr5s2
S
subshellin
O-Shellis
Full
3rd Gr
In
Kr4d!%5s2
Sp!
d-
subshell
in N-
Shell is
fullp-
subshellin
O-Shell
4h Gr
Sn
Kr4d!5s2
Sp?
d-
subshell
in N-
Shell is
full p-
subshellin
O-Shell
5th Gr
Sb
Kr4d!%5s2
Sp?
d-
subshell
in N-
Shell is
fullp-
subshellin
O-Shell
6'h Gr
Te
Kr4d!5s2
Sp*
d-
subshell
in N-
Shell is
fullp-
subshellin
O-Shell
T Y,Zr,Nb,Mo,Tc,Ru,Rh,Pd,Ag,Cd are the d-Block Transition elements which are there due to belated filling up of
d-orbitals in N-Shell which has 10 electron states permissible from 4d! to 4d!°.
For n=6, | can be 0, 1,2, 3, 4 and 5. This marks the Sixth Period.
Here we have K Shell , L-Shell , M-Shell are filled up and N-Shell and O-Shell is partially filled up. K,L, M full
and N(partially) and O(partially) corresponds to Xe. Partially filled N-Shell and partially filled O-Shell is in the
process of getting filled along with empty P-Shell up as shown in Table 2.11.
Table 2.11. Elements of SIXTH Period.
FifthPeriod 15 Gr 2nd Gr
Elements Cs Ba
Xe
corresponds
to K, L,M-
Shell filled
up.N-Shell
and O-Shell
partially
filled up.
Xe6s! Xe6s2
3rd Gr
Th
Xe4f!45qGs2
6p!
4h Gr
Pb
Xe4f!45d 652
6p?
5th Gr
Xe4f!45dGs2
6p?
FifthPeriod = 1 Gr 2nd Gr 3" Gr 4" Gr 5 Gr
P-Shell s-
orbital Is
first filled.T f-subshell in f-subshell in f-subshell in
Then N- N-Shell is N-Shell is N-Shell is
Shell is s- es full. The d- full. The d- full. The d-
filled up subshellin sabehellin orbital in O- orbital in O- orbital in O-
and then O- P-Shell P-Shellis Shell is full Shell is full Shell is full
Shell is getting Full andp-orbitalin andp-orbitalin andp-orbitalin
being filled filled P-Shell has P-Shell has P-Shell has
up. Then P- started filling started filling started filling
Shell p- up up up
orbital is
filled up.
tTLa,Hf,Ta,W,Re,Os,Ir,Pt,Au,Hg are the d-Block Transition elements which are there due to belated filling up of d-
orbitals in O-Shell which has 10 electron states permissible from 5d! to 5d!°.
Ce,Pr,Nd,Pm,Sm,Eu,Gd,Tb,Dy,Ho,Er,Tm, Yb,Lu are f-Block Transition elements which are due to belated filling
up of N-Shell f-orbital belated fillings from 4f! to 4f!4 .
Section 2.3.1.Applications of Noble Gases
Argon is used in glass chambers to provide inert atmosphere as for instance in Incandescent Lamp and in Siemen’s
Reactor. Helium is used as breather gas. He-Oy is used as breathing gas for deep-sea divers at a depth of 55m. This
prevents oxygen toxemia. This also prevents Nitrogen narcosis. Helium gases have replaced highly inflammable
Hydrogen gases in lighter than air applications.
Noble gases have multiple stable isotopes except Radon which is radio-active. Radon has a half-lifetime of
3.8days. It decays to Helium and Polonium which further decays to lead.
In each PERIOD of the periodic table, Noble gas has the highest first Ionization Energy and the Group I alkali
metal has the lowest Ionization energy. There is only weak Vander Waal’s force acting between Noble gas atoms.
Hence they have very low Melting Point and Boiling Point.
Noble Gases are nearly Ideal Gases and their deviations from Ideal Gas Law give important clues regarding the
inter-atomic distances of the gas atoms.
If an atom has only a under-populated sub-shell, it will tend to lose them to become positively ionized as it
happens in alkali metal. When metal atoms bond, a metallic bond occurs. When an atom has a nearly full electron
sub-shell, it will try to find electrons from another atom so that it can fill its outer sub-shell and become electro-
negative. These elements are usually described as nonmetals. The bond between two nonmetal atoms is usually a
covalent bond. Whereas metal and nonmetal atom come together and an ionic bond occurs. There are also other,
less common, types of bond but the details are beyond the scope of this material. On the next few pages, the
Metallic, Covalent and Ionic bonds will be covered in detail.
Chapter 2. Section 2.3.2. Application of Inert Gases.
Chapter 2, Section 2.3.2 gives the application of Noble Gases and Section
2.4 gives the three primary bonding namely Ionic Bonding, Co-valent
Bonding and metallic bonding and one secondary bonding namely Van-der-
Waal's weak force bonding.
Chapter 2. Section 2.3.2. Application of Inert Gases.
Helium- Helium is used as a component of breathing gases due to its low
solubility in fluids or lipids. For example, gases are absorbed by the blood
and body tissues when under pressure during scuba diving. Because of its
reduced solubility, little helium is taken into cell membranes, when it
replaces part of the breathing mixture, helium causes a decrease in the
narcotic effect of the gas at far depths. The reduced amount of dissolved
gas in the body means fewer gas bubbles form decreasing the pressure of
the ascent. Helium and Argon are used to shield welding arcs and the
surrounding base metal from the atmosphere during welding.
Helium is used in very low temperature cryogenics, particularly for
maintaining superconductors at a very low temperature. Superconductivity
is useful for creating very strong magnetic fields. Helium is also the most
common carrier gas in gas chromatography.
Neon- Neon is used for many applications that we see in daily life. For
examples: Neon lights, fog lights, TV cine-scopes, lasers, voltage
detectors, luminous warnings and also advertising signs. The most popular
applications of Neon would be the Neon tubes that we see for advertisement
or elaborate decorations. These neon tubes consist with neon and helium or
argon under low pressure submitted to electrical discharges. The color of
emitted light shown is dependent on the composition of the gaseous mixture
and also with the color of the glass of the tube. Pure Neon within a colorless
tube can obtain a red light, which reflects a blue shine. These reflected light
are also known as fluorescent light.
Argon- Argon is used for a diverse group of applications in the growing
industries of : electronics, lighting, glass, and metal fabrications. Argon is
used in electronics to provide a protective heat transfer medium for ultra-
pure semiconductors from silicon crystals and for growing germanium.
Argon can also fill fluorescent and incandescent light bulbs; creating the
blue light found in neon type lamps. By utilizing argon's low thermal
conductivity, window manufacturers can provide a gas barrier needed to
produce double-pane insulated windows. This insulation barrier improves
the windows' energy efficiency. Argon also creates an inert gas shield
during welding; to flush out melted metals to eliminate porosity in casting;
and to provide an oxygen-and-nitrogen free environment for annealing and
rolling metals and alloys.
Krypton- Just like argon, krypton can be found in energy efficient windows.
It is also found in fuel sources, lasers and headlights. It is estimated that
30% of energy efficient windows sold in Germany and England are filled
with krypton; approximately 1.8 liters of krypton. Being more thermally
efficient, krypton is sometimes chosen over argon as a choice for insulation.
Krypton can be found in lasers which works as a control for a desired optic
wavelength. It is usually mixed with a halogen (most likely fluorine) to
produce typically called "excimer" lasers. Krypton is sometimes used
within halogen sealed beam headlights. These headlights produces up to
double the light output of standard headlights for a brighter gleam. Also,
Krypton is used for high performance light bulbs which have higher color
temperatures and efficiency because it reduces the rate of evaporation of the
filament.
Xenon- Xenon is used for various applications. From incandescent lighting,
to development in x-rays, to plasma display panels (PDPs) and much more.
Incandescent lighting uses Xenon because less energy can be used to
produce the same amount of light output as a normal incandescent lamp.
Xenon has also made it possible to obtain better x-rays with reduced
amounts of radiation. When mixed with oxygen, it can enhance the contrast
in CT imaging. The revolutionize the health care industries. Plasma display
panels (PDPs) using xenon as one of the fill gases may one day replace the
large picture tube in television and computer screens. This promises a
revolution in the television and computer industries.
Nuclear Fission products may include a couple of radioactive isotopes of
xenon, which also absorb neutrons in nuclear reactor cores. The formation
and elimination of radioactive xenon decay products can be a factor in
nuclear reactor control.
Radon- Radon has been said to be the second most frequent cause of lung
cancer, after cigarette smoking. However, it can be found in various
beneficial applications as well. For examples through: radiotherapy, relief
from arthritis, and bathing. In radiotherapy, radon has been used in
implantable seeds, made of glass or gold, primarily used to treat cancers.
For arthritis, its been said that exposure to radon mitigates auto-immune
diseases such as arthritis. Those who have arthritis have actually sought
limited exposure to radioactive mine water and radon to relief their pain.
However, radon has nevertheless found to induce beneficial long-terms
effect. Some places actually have "Radon Spas". For examples: Bad
Gastem, Austria and Japanese Onsen in Misasa, Tottori. "Radon Spa" is a
relieving therapy where people sit for minutes to hours in a high-radon
atmosphere, believing that low doses of radiation will boost up their energy.
Section 2.4. Covalent bonding, ionic bonding, metallic bonding and
Van-der-Wal’s Weak Force bonding.
In this Universe all subsystems and this Universe itself is always moving to
minimum energy configuration because this is the stable equilibrium
condition. So is the case with atomic configurations known as molecules or
with atomic periodic configuration in a single crystal solid. How the atoms,
identical or dissimilar, will stably configure will be decided by the minima
energy configuration. Minima energy configuration will decide the type of
bond in a given molecule or in a solid state crystalline structure.
Covalent, Ionic and Metallic bonds are primary bonds and require more
than 1eV/atom dissociation energy.
Van-der-Waal’s Weak Force Bonding is secondary bonding and requires
less than 0.1eV/atom dissociation energy.
2.4.1. Ionic Bonding.
In a chemical compound of first group Alkali metal (Li,Na,K,Rb,Cs) and
seventh group Halogen element (F,Cl,Br,I) minimum energy configuration
is obtained through ionic bonded alkahalide salts namely KF, LiF,MgO,
CsCl and ZnS. Figure 2.5 illustrates the minimum energy configuration of
NaCl .
} Na’ + ci
} ) 0.236nm a(nm) ———~>
-4.26eV
Figure 2.5. Potential Energy vs Interatomic
Distance in NaCl molecule
1.In ionic bonded crystalline solids, the electrons are tightly bound to both
cations and anions nucleii hence these solids are good insulators.
2.Jonic crystals donot absorb light hence they are transparent to light.
3.Ionic salts are hard owing to strong ionic bonds,
4.These have high M.P.
5.These are brittle. One layer cannot slip past the other. They tend to cleave
and not deform.
6.Ionic salts are good IR absorbers because ionic salts electron-cloud
system has natural frequency or resonance frequency at IR. Therefore IR
are strongly absorbed by ionic salts.
7.These are non-directional and hence isotropic.
NaCl has fcc structure withg coordination number of 6. CsCl has bec
structure with coordination number of 8. In water, because of very high DC
relative permittivity (€, = 70), ionic salt is very soluable and cations and
anions easily dissociate and NaCl aqueous solution is a strong electrolytic
solution and will allow electricity to flow through. But Ionic salts are
insoluable in Covalent organic solvents such as Benzene(C6H6) and
Carbon Tetrachloride(CCl4).This is because the organic solvent has very
low dielectric constant therefore Ionic salts do not dissociate hence do not
dissolve in organic solvent.
2.4.2. Covalent Bond.
In ionic bond there is 100% transfer of electrons from Alkali metals to
Halogen elements, in the process both fulfill their octave as well they create
strong electrostatic attractive force. But in Silicon, Germanium or Diamond
the four electrons are shared with four neighbouring atoms in tetrahedral
crystalline structure which leads to complete fulfillment of octave for all
atoms except those on the boundaries which have incomplete bonds or
dangling bonds. This is 100% covalent bond and occurs only for Group IV
elements.[CHy,SiC,F>, Cl, Ho]
Group III and Group V, lead to 75% co-valent and 25% ionic.[GaAs]
Group II and Group VI, lead to 25% co-valent and 75% ionic.
Group I and Group VII, lead to 100% ionic.
Materials, elements or compounds, having 100% covalent boding has the
following properties:
1.Higly directional bond therefore anisotropic.
2.It can be non-polar as in Si, Ge and Diamond or it can be Polar as in
Water molecule.
3.Good insulator.
4.These solids are brittle or it can be soft also. If brittle they are transparent
and they cleave.
5.Low latent heat of fusion and evaporation.
The covalent bonds between atoms in a given molecule are very strong, as
strong as ionic bonds. However, unlike ionic bonds, there is a limit to the
number of covalent bonds to other atoms that a given atom can form. For
example, carbon can make four bonds - not more. Oxygen can form two
bonds. As a result, once each atom has made all the bonds it can make, as in
all the molecules shown above, the atoms can no longer interact with other
ones. For this reason, two covalent molecules barely stick together. Light
molecules are therefore gases, such as methane or ethane, above, hydrogen,
Hp, nitrogen, N> (the main component of the air we breathe, etc. Heavier
molecules, such as e.g. the isooctane molecule, are liquids at room
temperature, and others still, such as cholesterol, are solids.
In Figure 2.6. we see that Hydrogen Molecule achieves minimum PE
configuration through sharing its lone electron as a pair. The dissociation
energy is 4.5eV.
H2 molecule Dissociation Energy 4.5eV
Potential Engergy Vs
HH Internuclear Distance
Between Two Atoms
Internuclear Distance (pm)
Figure 2.6. Potential Energy vs Internumclear distance
alt.
Section 2.4.3. Metallic Bond.
Group I and Group II form strong metallic bond whereas Gr III form weak
metallic bonds. The conducting electrons are weakly held by their host
atoms. Therefore at room temperature they easily become delocalized and
belong to the whole lattice. This sea of delocalized electrons keep the (+)ve
charged lattice centers together. The metallic bonds are non-directional and
allow ionic centers to go past each other. Hence they are ductile, malleable
and have metallic luster. In Figure 2.7, metallic bond is illustrated. This sea
of de-localized electrons give high electrical conductivity and high thermal
conductivity attributes to metals.
B -90000000000
- 090000000000
O
O
-O00000000090
- 00000000090
+
+
+
+
+
+
© HQQQ00000900
E@O =~
A
Figure 2.7. Illustration of Metallic Bond. The positively charged lattice
centers are submerged in a sea of delocalized conduction band
electrons. These delocalized electrons allow high thermal
conductivity as well as high electrical conductivity.
Section 2.4.4.The secondary or Van-der-Waal’s weak force bonding.
This is Bond-induced bonding. Noble Gases are chemically inert are
chemically inert and consist of monoatomic molecule. But still have an
attractive force because of which they liquefy when cooled.
In absence of the three primary bonds, Van-der-Waal Bonding provides the
fourth kind of bonding mechanism which provides the cohesive force for
liquefaction and solidification. Macroscopic behavior such as Surface
Tension, Friction, Viscosity, Adhesion and Cohesion have their physical
basis in this mechanism.
Here there is no sharing of electrons or physical transfer of electron still
because of distortion of the electron cloud there may be a fluctuating dipole
or they may be a permanent dipole. In either case a weak attraction is
created among the molecules and this becomes the physical basis for Van-
der-Waal’s weak force bonding as shown in Figure 2.8.
In non-polar molecules it is by induction or by electronic polarization due
to close proximity. It is polarized and synchronized so that it is always
attractive as shown in Figure 2.8.In polar substances dipole to dipole creates
the electrostatic attraction.
It is this force which crystallizes Argon Crystals at -187°C and liquefies He
at 4K.
Arises from interaction between dipoles
¢ Fluctuating dipoles .
asymmetric electron ex: liquid H2
x clouds \, Ho > < H2
CD><Gep
second
bonding bonding
¢ Permanent dipoles-molecule induced
; secondary
- general case: a.» bonding <@(- )
. secondary
liquid HCI (cI) bonding <@(cl)
Figure 2.8. Fluctuating dipoles or permanent dipoles are the basis for
weak Van-der-Waal bonding.
Van-der-Waal force is much weaker than co-valent bond, ionic bond and
metal bond hence where this secondary bond is at work M.P is very low as
is evident from Table 2.12.
Table 2.12. M.P. of Solid Argon, Solid H and Solid Methane where
secondary bonding is at play.
Melting Bond Strength
Material Point (eV/molecule)
Solid Argon -189°C 0.08
Solid a
Aydrosen -259°C 0.01
Solid Methane -183°C 0.1
Section 2.4.5. Anomalous behavoiur of Water-Ice.
Animal-life is able to survive in a frozen lake essentially because of the
anomalous behavior of Water below 4°C. Below 4°C, the cooling should
make water denser but instead it makes water lighter as seen in Figure 2.10
as a result there is inverse Temperature Gradient as shown in Figure 2.11
and warmer water remains at lower layers well insulated from freezing cold
and provides favourable habitat for the marine world even though the
surface of the lake is frozen. This anomalous behavior occurs due to the
transition from Close-Pack Structure of Water to Open Pack Structure of Ice
as seen in Figure 2.9.
Loose hydrogen bonds between Strong stable hydrogen bonds
continuously moving HzO molecules between HzO molecules at 0 °C,
at 10 °C forming a rigid hexagonal crystal
lattice structure
H20 molecule
ate .
Large gaps between
molecules held rigidly
apart Lot of open space
3D lattice structure of ordinary ice in a refrigerator
www.goalfinder.com
Figure 2.9. Close-Pack Tetrahedral Structure of Water Molecule above 4degree C.
Open-pack hexagonal channel-like crystalline structure of ICE below Odegree C.
Anolmnous expansion of 1cc water between
4deg C and Odeg C due to phase transition
from CLOSE PACK to OPEN PACK
crystalline structure.
liquid
freezing T:
melting 4:
density maximum
42°C
volume in cr® of 1.00 g0f HO
temperature in °C
Figure 2.10. Anolmous behaviour of Water
below 4deg C.
Figure 2.11. Inverse Temperature Gradient
below 4deg C in a freezing lake.
Chapter 2. Solid State of Matter. Section.2.5. Bohr Model of Hydrogen Atom.
Chapter 2. Section 2.5 gives the success of Bohr's Model in explaining the orbital radii and energy associated with
the ground and excited states of Hydrogen Atom but it has problems dealing with more complex atoms.
Chapter 2. Solid State of Matter.
Section.2.5. Bohr Model of Hydrogen Atom.
According to Maxwell’s Equation an accelerated electron must give off radiation and lose energy. An electron in
an atom in a circular orbit is continuously accelerated towards the center. The acceleration towards the center is:
2
a=— wherev
Tr
= tangential velocity of electron,tangential to the circle of orbital rotation
and r = radius of the orbit. 2.1
This centripetal force is supplied by the electrostatic force of attraction by the nucleus of the Atom given by the
following expression:
v qx Zq
r 5 where Z = Atomic Number = number of electrons
r TE gr
= number of protons,4m comes because in Rationalzed MKS Units 1 Coulomb of charge
gives 1 Coulomb of flux hence at a distance r the electric flux density D = ri . r
ar
q = |chargelon an electron = |charge|on proton 22
Therefore:
xZ
m,v? = —— 2.3.
4megr
A circularly orbiting electron is continuously being accelerated towards the center hence it should be giving of
SYNCHROTRON Radiation. Synchrotron radiation has the same frequency as the rate at which electron is
orbiting the nucleus of the Atom. Infact this principle is being utilized in modem day Synchrotrons and Betatrons.
If this is to occur then electron must spiral into the nucleus and atom must collapse. But all around us the atoms are
stable. Hence Bohr made the following Postulates:
Postulate 1. Electrons are in Stationary States. There are some discrete energy states in which electrons are
permitted to stay.
Postulate 2. In the stationary states:
v h
Angular Momentum =I X w =m,r* X — =m, vr = Integral Multiple of a =nh 24
r 0
In a few years de Broglie in France gave his Hy pete of matter wave and that:
h
Wavelength of the matter wave of electron=i= : = aa 2.5
Combining Bohr’s Law (2.4) and de Broglie hypothesis (2.5) we get one,
2nr = circumference of the orbit = nA
Equation (2.6) implies that only the electrons which form a standing wave along their respective orbital paths as
shown in Figure 2.12. are permitted to stay in stationary orbits. If the electron does not form a standing wave it
starts radiating and it spirals in as shown in Figure 2.13.
Figure 2.12. Electron's matter wave forms a
standing wave corresponding to every
principle Quantum Number.
Figure 2.13. Electron in a radiative orbit
spirals inas shown and atom collapses.
From (2.4) squaring both sides we get:
(nh)?
v= 2.7
(m,r)?
Multiplying both sides by the mass of electron we get:
2 — (nh)? 28
mv = a WH at
Both (2.3) and (2.8) give twice the Kinetic Energy of the orbiting electron. In (2.3) the KE has been obtained from
Newtonian Mechanics consideration whereas from Bohr’s postulate (2.8) has been obtained. But the two are equal.
Based on this equality we obtain:
qXZq _ (nh)?
= 29.
4neyr m,(r)?
Rearranging the terms we obtain the radius of the orbits as follows:
7, (radius of the n™ Orbit around an atom of Atomic Number Z)
4 bal 2 2 h 2 2 h 2
Bic! EM fal PJ al, ee ee ee oe
m,q°Z Z} mm,q* Z mm, q"
Calculating the Bohr Radius we get : ag= 52.9459 picometer(pm).
Here we have taken the Universal Constants and electron charge and mass as follows:
q =charge of electron = 1.6 x 10°1°C,m, = rest mass of electron = 9.1X 10-*'Kg,
Planck'sConstant = h = 6.62 X10-“J —s, £,(abs.permittivity) = F/m.
36m X 10°
According to Quantum Mechanics, the maximum probability density occurs at Bohr Radius but the total
probability within a sphere of Bohr Radius is less than 65%.
Section 2.5.1. Total Energy of Electron in n th Orbit of an Atom of Atomic Number Z.
Total Energy TE is given as follows:
2.11
i728)
1
TE = Kinetic Energy (5,07) + Potential Energy (-
2 Ame gr
From (2.3) we find that 2xKE = magnitude of PE hence TE simplifies to:
1/ qxXZq qx Zq
T= =(- ) Se
A4megr 8reqr
2.12
2
Substituting the nth Orbit Radius (r,) as given in (2.10) we get TE as follows:
epics SOOO, 8 Zz? x(=)x fail 2.13
Brey (=) eis n? ay Brey :
Z 0
From (2.10) and (2.13) gives the orbital radius and the corresponding energy for any atom of Atomic Number Z.
Section 2.5.2. Calculation of the orbital radii and Total Energy of different principle Orbits of Hydrogen
Atom.
We use (2.10) and (2.13) to determine the orbital radii and total energy of the discrete energy states of Hydrogen.
Here Z=1 is taken since H atom has 1 electron and 1 proton. In (2.13) the results are obtained in Joules. To obtain
results in eV we must divide Joules by (qCx1V). The results of calculations is given in Table 2.13.
Table 2.13.Radii and total Energy of Hydrogen Atom Principal electronic Orbits.
(Principal Quantum Number)n r, (Angstrom) Total Energy(eV)
1 0.529 -13.6
2 2.12 -3.4
3 4.76 -1.5
4 8.47 -0.85
Based on the above Table a correct theoretical basis of the Spectral Lines of Hydrogen Atom was provided by Neil
Bohr as explained in Figure 2.14.
0
-0.85eV <—
4.511eV mm
series
-3.4eV oe
Transition from higher states
to n=1 gives Lyman Series
Transition from higher states
to n=2 gives Balmer Series
Lyman
series
Figure 2.14. Theoretical Basis of the Spectral Lines
observed in Hydrogen Atom.
As seen in Figure 2.14:
Transition from higher states to ground state namely n =1 gives Lyman Series.
Transition from higher states to next excited state namely n =2 gives Balmer Series.
Transition from higher states to third excited state namely n =3 gives Paschen Series.
Transition from higher states to fourth excited state namely n =4 gives Brackett Series.
Transition from higher states to fifth excited state namely n =5 gives Pfund Series.
Section 2.5.3. Problems with Bohr Theory.
Bohr’s Model is unable to show that apart from Principal Quantum Numbers(n) there are azimuthial/orbital
quantum number (/ ) , magnetic quantum numbers (m) and spin quantum numbers (s).
In 1925, Erwin Schrodinger proposed Wave Mechanics applicable to matter wave.
His proposition was the following:
Hy =Ey 2.14
H is the Hamiltonian Operator which includes KE plus PE operator. The Hamiltonian Operator operating on
Matter Wave yields the Eigen Values of Hamiltonian Operation.
This equation was applied to an electron orbiting a proton in a Hydrogen atom. Since it had spherical symmetry
hence Spherical Coordinates or Spherical Frame of Reference was adopted. The solution of Schrodinger Equation
yielded the three non-relativistic quantum numbers namely:
Principal Quantum Number ‘n’ which gives the quantization of Energy of Orbital Electrons.
Azimuthial/Orbital Quantum Number ‘I’ which gives the quantization of the orbital angular momentum L.
Magnetic Quantum Number ‘m’ which gives the quantization of the orientation of L with a special frame of
reference Z axis.
(2.14) is non-relativistic equation hence only non-relativistic Quantum Numbers are predicted.
Relativistic treatment yields the Spin Quantum Numbers.
These four Quantum Number have been explained Section 2.4.
Bohr Theory could not be used to determine energies of atoms with more than one electron. It was unable to
explain fine structure observed in H atom spectra. It cannot be used to understand bonding in molecules, nor can it
be used to calculate energies of even the simplest molecules. Bohr’s model based on classical mechanics, used a
quantization restriction on a classical model.
Section 2.5.3. Calculation of Outermost Orbital Radii and Total Energy associated with Outermost Electron
in Noble Gases He, Ne, Ar, Kr, Xe and Rn.
Equations (2.10) and (2.13) cannot be directly applied to atoms heavier than Hydrogen. This is because in complex
atoms while studying the outermost orbit we have to account for the screening effect of the intervening electron
cloud.
The Inert Gases and their electronic configurations are tabulated in Table 2.3
Table 2.3. Shell Structure of Inert Gas Atoms.
K- Tie M- O- p-
Gas Z Shell(n=1) Shell(n=2) Shell(n=3) N-Shell(n=4) Shell(n=5) Shell(n=6)
He 2 1s2
Ne 10 1s? 287 .2p°
Ar 18 1s2 2s? ,2p® 3s ,3p®
Kr 36 Is2 2s? 2p 3s" 4s? ,4p®
»<P 3p%,3d! sD
2
2 2 576 3s 2 4n6 4ql0 2 o6
Xe 54 1s 2s° ,2p '3p63d!° As* ,Ap’,4d 5S* ,op
382 As? 582
2 2 576 2 eno
Rn 86 Is 2s° ,2p (3p8,3d!0 Aps,4d!,4f!4 psa! 6s* ,6p
From Table 2.3 it is evident that in Neon, K-shell electrons and L Shell s-orbital electrons will have a screening
effect to the extent that p-orbital in L-Shell will experience only the pull of (10-4) protons and screening factor (S)
is 4.
In Argon, K-shell , L-Shell and M-Shells s-orbital electrons will have a screening effect to the extent that p-orbital
in M-Shell will experience only the pull of (18-12) protons and screening factor (S) is 12.
In Krypton, K- shell, L-Shell, M-Shell and N-Shell s-orbital electrons will have a screening effect to the extent that
p-orbital in N-Shell will experience only the pull of (36-30) protons and screening factor (S) is 30.
In Xeon, K-shell, L Shell, M-Shell, N-Shell and O-Shell s-orbital electrons will have a screening effect to the
extent that p-orbital in O-Shell will experience only the pull of (54-48) protons and screening factor (S) is 48.
In Radon, K shell, L Shell, M-Shell, N-Shell, O-Shell and P-Shell s-orbital electrons will have a screening effect to
the extent that p-orbital in P-Shell will experience only the pull of (86-80) protons and screening factor (S) is 80.
In 1964, J.C.Slater empirically measured the covalent bonds of all the elements and published their outer orbital
radii.by the title “Atomic Radii in Crystals” in Journal of Chemical Physics, Volume 41, No.10, pp 3199-3205.
I used these empirical values of atomic radii and the modified formula for (2.10) namely:
1, (radius of the n** Orbit around an atom of Atomic Number Z)
nt
= X dy 2.15
Z-S
Using the empirical values of the radii and using (2.15) I determined the actual screening effect in each case.
The magnitude of the total energy of the outermost orbital electron directly gives the First Ionization Energy and
this has been experimentally measured by .
Table 2.4.Empirical Radii(R), Z, calculated S* , experimentally measured First Ionization Energy of the 6
Inert Gases.
Inert n(P.Q.N of outermost First Ionization
Gas R(pm) Z S* orbit) Energy(eV)tT
He 31 2 0.292068 1 24.8
Ne 38 10 4.42675 2 21.5
Ar 71 18 11.2885 3 16.0
Kr 88 36 26.3735 4 14.0
Xe 108 54 41.744 5 12.0
Rn 120 86 70.1162 6 11.0
S*- calculated value of Screening effect by me.
E*- calculated value of total energy of the outermost orbital electron.
P.Q.N- Principal Quantum Number.
TExperimentally determined by Greenwood(reference not available)
Tutorial on Chapter 2: Insulator, Semi-conductor and Metal.
This module gives a set of problems on Band Theory of Solids.
Tutorial on Chapter 2: Insulator, Semi-conductor and Metal.
1. Q.1. Fermi-level in Metal :
h? 2
E, ==— X (—)3 where n= density of conduction electrons in the metal h
2m, 81
= 6.62606876 x 10° **J —s
Determine the Fermi-level of Sodium and Copper where n =
2.5x102electrons/cc and n = 8.5x10?electrons/cc.
Wr Light absorbed
Metals Er(eV) n(#/cc) (eV) and re-emitted
Li 4.72
Na Bel2 2.5x107 2.3
K 2.14 - Pee:
Rb 1.82 -
Cs 1.53 1.8
Cu 7.04 8.51074 2 Orange
Wr Light absorbed
Metals Ey(eV) n(#/cc) (eV) and re-emitted
Appears White
- = 4 | thevisile
spectrum
Au 11.7 23 Yellow
Ca 32
Ba yas
Pt 5.3
Ta 4.2
w 4.5
Most suitable as
StBaCuO 1 Cathode material
in Thermoionic
Tubes.
1. Q.2.Density of States [N(E)] is defined as number of permissible states
per unit volume per unit energy level:
V2m2/? E42
N(E)=
n*(a)*
-
’
Show that mean energy of electron in conduction band is (3/5)Ep .
1. Q.3. Determine P(E + kT) in Fermi-Dirac Statistics. Determine
Temperature T at which P(E, + 0.5eV) = 1%.[Answers: 0.27,
1262Kel]
2. Q.4. Determine the density of occupied states at (Ep + kT) at T=300K.
Find the energy E below (Ef ) which will yield the same density of
occupied states. [Answer 0.92eV]
AE = 0.92eV
(E)——> PE) —> N(E)P(E) ——>
Density of States Probability of Occupancy Density of Occupied States
1. Q.5.Determine the Fermi Velocity of conduction electron in Metal
Copper and Thermal Velocity of conduction electron or hole in Semi-
conductor.
[ Hint:
1 2_3 : ‘ 1 2 =
In Metal >m,v; = —E, and in Semi — conductor 3 MeV tnerma = 5 kT:
]
{Answer : Fermi Velocity in Metal Copper is 2.7256x10°m/s and in semi-
conductor Thermal Velocity is 1.17x10°m/s]
1. Q.6.In Copper, p(resisitivity)=1.67*10°Q-m and o(conductivity)=
6x10’S/m = qun. Where n=8.5*108electrons/m*. Determine
conduction electron mobility. [ 40cm?/(V-s)]
2.Q.7. 3m long Copper wire of R= 0.03 and the current I through the
wire is 15A.
a. Determine Voltage Drop across the Wire.[0.45V]
b. Determine the Electric Field within the Wire.[0.15V/m]
c. Determine the drift velocity of electron within the wire.[0.6x10°m/s]
d. Determine the Fermi Velocity.[2.7256x10°m/s]
e. Determine the Mean Free Time or Relaxation Time.[2.27x10"4s]
. Determine the Mean Free Path and compare with the lattice parameter
of Copper.[618A°]
Lae}
1. Q.8. Typical Resistivity of N-Type Silicon is 1 Q-cm and typical
current density is 100A/cm?. Calculate the applied Electric Field in
Silicon Sample and calculate the drift velocity. Assume mobility of
electron to be 1400cm?/(V-s) [Answer:100V/cm and 140x10°cm/s]
2. Q.9. Calculate the intrinsic resistivity of Ge, Si and GaAs. [107Q-cm;
3.4x10°Q-cm; 7x10°Q-cm].
3. Q.10. In all the calculations above we have assumed that the effective
mass of electron or hole within the solid is the same whereas it is not
true. The effective mass in Metals are generally higher than the free
Space mass and in semiconductors it is lighter as seen from the Table
below.
Material Si Ge GaAs InAs AlAs
m,/Me 0.26 012 0.068 0.023 0.88
™ hole/Me 0.39 0.3 0.5 0.3 0.3
E,(eV) 142 0.67 1.42 0.35 22
Electrons and holes in a crystal interact with the periodic potential field in
the crystal. They surf over the periodic potential variation of the crystal
developing roller coaster effect which leads to drastic reduction in effective
mass. If effective mass is considered the thermal velocity comes to be much
higher.
For electron with effective mass of 0.26m, the thermal velocity is
2.3x10°m/s.
For hole with effective mass of 0.39m, the thermal velocity is 2.2x10°m/s.
1. Q.11.Find the drift velocity, mean free time and mean free path in a P-
Type Sample with hole mobility = 470cm?/(V-s) and applied Electric
Field E= 10°V/cm. All calculations will have to be carried out in
Rationalized MKS units that is mobility will be taken as 0.047m2/(V-
s). Thermal Velocity at Room Temperature is approximately 10°m/s.[
4.7x10°cm/s, t =0.1ps, MFP = 100A°]
SSPD_Chapter 1_Part 12_Quantum Mechanical Interpretation of
Resistance
SSPD_Chapter 1_Part 12 gives the quantum mechanical interpretation of
Resistance in conductors.It is not the lattice parameter but the mean free
path which decides the resistance.
SSPD_Chapter 1_Part 12_ Quantum Mechanical Interpretation of
Resistance in a conductor.
1.12. SCATTERING OF CONDUCTING ELECTRONS AND
RESISTIVITY OF SOLIDS.
Here we are dealing with a 3-D orderly configuration of atoms arranged as
a single crystal of infinite dimensions. We have already seen that electron
has all the properties of light wave namely: Bragg reflection, refraction ,
interference and diffraction. Therefore scattering of electrons cannot be
understood in a classical manner.
Liquid state of matter causes fifty times more scattering of electron as
compared to the gaseous state of matter where as liquid is thousand times
more dense. Therefore extent of scattering does not depend on the density
of scattering centers but on the disorderly arrangement of the scattering
centers. Here it will be proper to point out as to exactly what the difference
is between scattering and reflection.
A smooth plane reflects light whereas a rough plane scatters light. In
reflection the angles of incidence are identical hence angles of reflection are
identical. As a result the total incident beam of light is reflected. In a rough
plane, the constituent rays of the light beam are reflected in different
directions since the angles of incidence are different for different rays. This
is known as scattered light. These two phenomenon are shown in
Figure(1.68).
Incident beam Reflected beam
eK
Refeciion from ac emooth surface
Figure 1.68.a. Reflection from a smooth surface keeps the reflected
rays parallel. Hence incident beam is parallel and reflected beam is
parallel.
NA
Figure 1.68.b. Parallel beam incident on an inregular and rough
surface causes scattering. The reflected rays are ho more
collmmated. They scatter out in all directions. Hence the reflected
beam is diffused in nature.
Figure 1.68. Reflection and Scattering.
Electron is a matter wave probability amplitude matter wave. In 3-D orderly
crystalline solid the propagation of electron causes the interaction with and
vibration of all the intervening lattice centers. The lattice centers become
the emitters of secondary wavelets.
The secondary wavelets interfere and propagate forward in the direction of
constructive interference.
a. If the crystal is at O Kelvin then there is no thermal vibration. Also
assume that there are no crystal defects and that there are no
impurities.
b. Also the crystalline lattice in not bounded by surfaces and is infinite in
the three directions.
c. Also electrons are in the outer partially filled conduction band well
removed from the upper edge of the conduction band.
If all these conditions are fulfilled then the direction of constructive
interference will be the direction of incidence and electron will propagate
forward as if there is no obstacle in its path. The first impulsive energy will
sustain the electron in a straight line propagation path with uniform velocity
with no dissipation of energy. This propagation will continue for infinite
time and till infinite distance. This is exactly as Newton had predicted about
the inertness of bodies: A body at rest will continue to be at rest and a body
in motion will continue to be in motion in a straight line with an uniform
velocity unless made to act otherwise by the application of force.
This is what is more popularly known as SUPERCONDUCTIVITY.
Near the upper edge of the conduction band, electrons will suffer Bragg
Reflection and this will be decided by the direction of orientation of the
lattice plane from where the specular reflection takes place.
Lattice thermal vibration or/and lattice defects or/and lattice impurities
cause the disorderliness and this disorderliness leads to change of direction
of propagation or the change of direction of constructive interference. This
change of direction is called scattering of electron. The change of direction
is random and the distance over which this happens is statistically varying .
The mean distance over which a straight line motion is maintained before
scattering occurs is called the mean free path and the mean time taken to
cover the mean free path is called mean-free time. The electron scattering
within a real crystalline lattice which suffers from all defects and
imperfections and which is at Room Temperature has been illustrated in
Figure(1.69).
e a =
Straight electron path
‘ w e
Figure 1.69.a. Electron is moving along a straight path in an
ideal crystalline lattice. An ideal crystalline lattice has no defects,
no imperfections and no thermal vibratins. Electron experiencesa
no scattering.
Black Circles are Lattice Centers arranged in a perfect orderly
pattern.
Figure 1.69.b. A real crystalline lattice is shown with lattice centers
randomly displaced with respect to orderly arrangement of lattice
centers due to theermal vibration. This perturbation in the orderly
alrangement leads to random motion of electron with no net
displacement in time. As we see that after eight scattering electron
initially starting from 'A' finally ends at 'B' which is is more or less in
the same location as ‘A’.
Black circles denote the orderly pattern of crystal.
Open circle reprent the perturbation in the regular orderly
alrangeint of the latice centers.
Figure 1.69. Scattering Phenomena of electron in a real crystalline lattice
where lattice thermal vibration, lattice defect, lattice impurities and lattice
boundaries are present.
This scattering of electron does not depend on the density of lattice centers
but on the degree of disorderliness of the lattice centers and this degree of
disorderliness depends on thermal vibration, lattice defects and lattice
impurity density.
By increasing the temperature, the amplitude of lattice thermal vibrations is
increased ;
During the growth of the crystal, lattice defects are increased;
During doping and diffusion, impurities are introduced in the crystal.
All these factors contribute towards disorderliness which in turn contribute
towards the scattering of the conduction electrons. It is this scattering which
creates resistance in a conducting solid. There are two parameters which
describe the scattering phenomena:
Mean free path(<I>) and mean free time(t).
As seen in Figure(1.69.b), at 1), lo, ls, ly,........ distances the direction
changes.
Therefore mean free path:
SD YE WW Nicastncd cacti saison tne vec seeaveadent twndnest 1.100
If the time taken is ty, to, ts, t4, .... along the randomly varying paths then
the mean free time:
= CIN Gases sewsaces San ntatea se scsue tai iste on cease 1.101
And <I>/t = average thermal velocity= v thermal ..................seeeeseees
1.102
By Equipartition Law of Energy, every degree of freedom has an average
thermal energy = (1/2)kT. Since conducting electron has three degrees of
freedom(x, y, z) hence the total average thermal energy of a conducting
electron is equal to (3/2)kT.
Therefore (1/2)m e * .v thermal 2 = (3/2)KT ............ccccccsccescccescceees
1.103
Where k= Boltzman’s constant=1.38x10-77J/Kelvin = 8.62x10°eV/Kelvin
And T is temperature at Kelvin scale. m, = effective mass of electron.
Temperature in Celsius scale is added to 273 to obtain temperature in
Kelvin Scale. The zero of Kelvin scale occurs at -273°Celsius. At this
temperature i.e. at 0 Kelvin the amplitude of thermal vibrations of lattice
centers is zero and if there is no lattice defect and no doping then we have a
perfect orderly lattice which will behave like a superconductor.
1.12.1. Quantum Mechanical basis of Resistance in a conducting Solid.
An ideal crystal with no lattice defects, no dopents and at zero Kelvin
temperature behaves like a superconductor.
In a normal metal, resistance is always present.
Let us consider a cylindrical metal of length L cm and A(cm)* cross-
sectional area. A potential difference of V volts is applied across it as
shown in Figure(1.70).
The electric field along the longitudinal axis is:
(a) The application of an electric field across cylindrical metal specimen
Figure 1.70.a. An electric field is applied along the positive z axis of a
cylinderical metal conductor. Electrons move along the negative z axis
causing an electric currentI=V/R
where R = (resistivity).(L)/A=R = pL/A according to Olum's Law.
Elecrons move along the negative z axis shown by the broken
line.
vdrift/average
t, te t;
t (Sec)
(b) The instantaneous drift velocity of electron with respect to time
Figure 1.70.b. Instantaneous drift velocity profile with respect to time.
t,
Figure 1.70. Electron drift in z direction(longitudinal axis of the
cylinder) after the application of the z-axis electric field.
According to Kinematics, the electric field is applied in z direction. This
will cause an acceleration in [-z] direction since electron is negative. This
acceleration will continue until the electron gets scattered. At the point of
scattering all the kinetic energy of the electron is imparted to the lattice and
the electron starts anew from zero velocity. Since this is an statistical
phenomena hence acceleration time periods are ty, ty, tz, ty,...... and the
terminal velocities are Vi1, Viz, Vi3s Vidsereeeeeeeeeeees Acceleration time
periods
re eg a © Reeere are the same as the transit periods along the randomly
changing straight line segments as defined in Eq.(1.100)
From kinematics: v,, = u+at,
where a = acceleration= F/m e * = ge/ me * = q(V/L)/ me *
sAWndaeWeubedbueekateckevenscensicedvs 1.105
Viteeressreeaetcceee hes: 1.106
Therefore the average drift velocity over n"
acceleration period is:
V n= (vtn + 0)/2 = (qe/ 2m e *) (C1 )..... cece e cece esse cece eees 1.107
Suppose during the flow of current from one end to the other, a given
electron undergoes N scattering. This means it undergoes N acceleration
periods.
The average drift velocity during N periods of acceleration which occur
during the flow of the current is:
Vdrift = [yivnl/N = [d(qe/ 2Me*) (t,)/N
Therefore Vapi = (Ge/ 2Me*)[>.(t,)J/N
v drift = (qe/ 2m e *)t = (q T/ 2M *).E oo... ccc cceeeccescccceccessceesceees
1.108
where Tt = [}‘(t,)]/N= mean free time as defined in Eq.(1.101) and
LU, = electron mobility = (q t/ 2m,*)
This relation was used to determine the electron mobility in Table 1.11, Part
ah;
Now under low drift velocity condition, drift velocity varies directly as the
applied field ¢ and the constant of proportionality is known as the drift
mobility(y1 (cm?/(v-sec)) as shown in Figure 1.72.
y drile=(g:1/ 2:6”) S10 Bis eececseieednpiaiessecewastobsseengesaeiices
1.109
Current density =J= number of Coulombs/second
Therefore J=-qxnumber of electrons flowing through unit cross-sectional
area per second.
If we assume that there is no diffusion of mobile carriers and we only have
electric drift of the mobile carriers then number of electrons flowing
through unit cross-sectional area per second=(1cm7)(Vgrif)(M)
where n= number of conducting electrons per unit (cm)*
and Varif, = drift velocity of electrons;
Therefore J= (q.v drift .n) Coulombs/(sec-cm 2 )
Uauslwicvensete Medes notuanuwevoesesceus 1.110
On T= GEE HO Ev caaccancccincae sole cewnascaaw tes iewswdisincenssabpases ssaeass
1.111
Where o(conductivity) = 1/p(resistivity)
J = q.p.€.n =0.€
Or J = I/A = (1/p)x(V/L)
Therefore V/I = R = (pL)/A> Ohm’s Law.
bactedussedzonneretneremsvesteddenee ees 1.112
In Eq.(1.109) , we find that drift mobility is directly dependent on the mean
free time between two consecutive scatterings. Mean free time (tT) is
dependent on scattering. Larger is the perturbation in the lattice network
from the ideal lattice more frequent will be the scattering and hence shorter
will be the mean free time. As the perturbation from the ideal condition or
the disorderliness is reduced, so will the scattering phenomena be reduced
and mean free time will become longer. Under ideal condition there will be
no scattering and mean free time will become infinite. Here the mobility
becomes infinite , conductivity becomes infinite and resistivity becomes
zero. This is a superconductor. Here once an electron gets an impulsive
push it continues to travel in a straight line for infinite distance and for
infinite time without any energy dissipation. Hence initial kinetic energy
imparted by the impulsive push is conserved forever by the conducting
electron. This is tantamount to a current flow in a close loop superconductor
without any battery connected to the circuit.
In a normal conductor even with no battery connected, the mobile carriers
are undergoing random motion with no net displacement. When an electric
field is applied then superimposed on this random motion there is a net
displacement of electrons in the opposite direction to the electric field. This
has been shown in Figure(1.71). The net displacement is given by AL in
time At.
Z axis
Thin line is the random motion with no electric field
hence there no net displacement.
If an electric field is applied in -z direction then the
scattering follows the bolder line and after four
scatterings the electron is displaced with respect to the
original position by by AL in time At in positive z direction
and the electric field is applied in -z direction.
Figure 1.71. The net drift experienced by an electron
under the influence of electric field.
As can be seen in Figure 1.71 electron is undergoing continuous random
motion under the influence of thermal energy. As temperature increases
electrons become more restless and start meandering along more zig zag
path but they never make a net displacement in any direction as shown by
‘thin line’ zig zag path. But as an electric field(¢) is applied in — z direction,
electron follows bolder line and as seen in Figure 1.71 it experiences a net
displacement of AL in +z direction in At.
AL/At = average drift velocity = Varift = Un€ 3
In 1911, Kamerling Ons detected superconductivity and superfluidity in
solid mercury at 4.3Kelvin.
In 1962 a Russian Scientist was awarded the Nobel Prize in Physics for his
study on superfluidity and superconductivity in liquid Helium at 4Kelvin.
In 1987 Karl Alex Muller and Bednorz of Germany were awarded the
Nobel Prize for discovering the superconductivity in ceramic Yettrium
Barium Copper Oxide [Y;Bay (CuO)s | at liquid Nitrogen temperature
77Kelvin. Bednorz was a research student under Muller at that time.
Before we leave this Chapter on mobility and resistance it will be
appropriate to introduce the reader to the concept of two kinds of mobilties:
First kind Uattice due to the scattering caused due to lattice thermal vibration
and it is temperature dependent;
Second kind Himpurity due to the scattering caused due to the dopents and/or
crystalline defects;
At liquid Helium temperature that is at less than 4Kelvin there is no
scattering due to thermal vibrations but scattering due to impurity and/or
crystal defect persist. Therefore semiconductors never become
superconductors. Even at OKelvin residual resistivity persists due to
impurity and/or crystal defect.
So the effective mobility is given by the following reciprocal relationship:
1/ pp =1/ p lattice + 1/ pr IMPUTity ............ceccccoescecccccscoesccccoecees
1.113
The quantum-mechanical model of electron scattering , which has been
presented in this chapter is valid only when drift velocity is much lower
than the thermal velocity. When we reach high Electric Field region, the
electric energy is directly transferred to the crystalline lattice and the drift
velocity saturates at Scatter Limited Velocity as shown in Figure(1.72). The
Scatter Limited Velocity is 10’ cm/sec for Silicon and it can be derived
from the following relation:
(1/2)(m e *)(v scatter limited ) 2 = (1/2)(m e *)(v thermal ) 2 =
CS) QT scecswseVolreeeas iiiewdestnniis 1.114
rT | a | Ts
E(Viem) — SOS
Figure 1.72. Vdrift versus Electric Field.
Figure 1.72. In high electric field region the drift velocity saturates at
scatter limited velocity.
The Quantum Mechanical perspective tells us that electron is not impeded
by the lattice centers. If the lattice centers are perfectly orderly at OKelvin
then electron if imparted an impulsive energy will acquire a finite kinetic
energy and with this KE it will continue to travel in a straight line through
the crystalline lattice till infinity. What impedes the flow and causes the loss
of KE is not lattice centers per se but the disorderliness of the lattice center
network. This is the reason why Graphene is working out to be a wonder
material with a very large drift mobility. Graphene is a sheet of orderly
arrangement of hexagonal structure which is unperturbed or unbroken over
large distances hence mean free path in Graphene is of the order of
micrometers as compared to the mean free path in GaAs where it is of the
order of a fraction of micrometer.
SSPD_Chapter 1_Part 11_Solid State of Matter
SSPD_Chapter1_ Part11 gives a elaborate classification of Solid State of matter. This first part of Part 11 attempts
to explain that electron's debroglie wavelength will decide how it interacts with solids. It is the difference in this
de-broglie wavelength which makes conducting electron behave so very different in metals and semi-
conductors.This part also describes how particle accelerators are gigantic microscopes which probe innermost
recesses of matter.
SSPD_Chapter 1_Part 11_Solid State of Matter
1.10.DEFINITION OF INSULATORS, SEMICONDUCTORS AND INSULATORS.
1.10.1. SIX STATES OF MATTER.
There are six states of matter: gas, liquid, solid, plasma, Bose-Einstein Condensate and Fermi-ionic Condensate.
Gas has no fixed shape or volume. They have the shape and volume according to the vessel they occupy. Inter
molecular distances are large and molecules are independent of one another. This state of matter obeys the Ideal
Gas Law.
Liquid has a fixed volume but no fixed shape. It takes the shape of the vessel it occupies. Intermolecular distances
are fixed and molecules experience cohesive force with respect to one another.
Solids have a fixed volume and fixed shape. The molecules are arranged in an orderly fashion giving rise to a
crystalline structure. Because of the variation in the range of orderliness , the crystalline structure is classified as
Single Crystal, Poly-crystal and Amorphous. The range of orderliness is shown in Figure(1.39).
For 300,000 years after the Big-Bang, temperature of the Universe was above 4000K and all matter was in
PLASMA STATE. This is a soup of electrons, protons and neutrons. As long as matter is in plasma state,
RADIATION dominates. Gravitational attraction is dominated by electromagnetic forces and thus the gravitational
accretion is prevented.. As soon as temperature falls below 4000K, plasma recombines to form neutral mass of
Hydrogen (70%), Helium(30%) and traces of Lithium. At this point radiation decouples and matter dominates. The
decoupled radiation carries the imprint or profile of the matter distribution through out the Universe at the time of
de-coupling. This decoupled radiation persists till today in almost its pristine state in which it decoupled from the
matter. This decoupled radiation is known as Cosmic Microwave Background Radiation (CMBR). The latest study
of CMB by WMAP show that indeed there are hot spots and cold spots in CMB implying that in the remote past
matter distribution did contain the unevenness which would eventually become the seeds for the formation of
clusters, galaxies, solar systems and planets.
Bose-Einstein Condensate and Fermi-ionic condensate are described in Section (1.15) . They are closely related
but Fermi-ionic Condensate formed at a lower temperature than the temperature at which Bose-Einstein
Condensate is formed. It manifests a complex of spectacular behavior:
i. It flows through tiny capillaries without experiencing any friction;
ii. It climbs in the form of film over the edge of vessels containing it. This phenomena is referred to as ‘film
creep’;
iii. It spouts in a spectacular way when heated under certain conditions;
iv. If it is contained in a rotating container, the content in He-II Phase never rotates along with the container.
1.10.2. SOLID STATE OF MATTER.
With the development of Quantum Mechanics, Band Theory of solid was proposed by Felix Bloch[Appendix
XXXXIII]. As already seen in the last section, electrons in a single crystal solid occupy energy bands separated by
forbidden zones known as Band Gaps as shown in Figure(1.40) , Figure(1.41) and in Figure(1.49).
The outer band is known as the conduction band and the band just below is valence band. In insulators and
semiconductors, the energy band gap between valence and conduction is wide of the order of eV and conduction
band is completely empty at low temperatures for both insulators and semiconductors. That is at 77K (liquid
Nitrogen temperature) and below, semi-conductor is an insulator. Only near room temperature and by introduction
of controlled amount of impurities that semiconductor acquires a certain degree of conductivity. As we will see in
semiconductor chapter the effective density of states at the lower edge of conduction band (Nc)
and at the upper edge of the valence band (Ny) is nearly the same as shown in Table(1.10). This is of the order of
10419 permissiblestates per centimeter cube. Hence as soon as doping approaches this order of magnitude, number
of conducting electrons are comparable to the available energy states which is the criteria for degenerate systems.
Hence at that order of magnitude of doping, semiconductor becomes degenerate and behaves like a semi-metal.
So an INSULATOR can be defined as the solid which has an empty conduction band and a large band gap , of the
order of 4eV or more. There electron- hole pair cannot be thermally generated . Hence it remains non-conducting
at all temperature.
On the contrary, SEMICONDUCTORS are non conducting and hence insulator below liquid Nitrogen temperature
and above liquid Nitrogen temperature they acquire conductivity either due to thermal generation of electron-hole
pair or due to contribution of conducting electrons by net donor atoms or due to holes contributed by net acceptor
atoms.
Metals have partially filled conduction band or overlapping conduction and valence bands. As a result, there are
mobile electrons available in copious amount. This amount is of the order 10? per centimeter cube. As we see in
the Table(1.10), atomic concentration in Solids are of the order of 10? per c.c. and each atom contributes an
electron for conduction. Hence availability of conduction electrons is 12 orders of magnitude greater than intrinsic
Silicon and 9 orders of magnitude greater than that of intrinsic Germanium. Because of this large number of
mobile carriers present in metal that resistivity is so much lower in metal than that in semiconductor. This point
will become clearer as we proceed with the quantum-mechanical interpretation of conducting electron in metal or
in semi-conductor.
As seen in Table(1.10) , the mobility of conducting electrons in semiconductors is much higher than that in metal
which is typically 44cm*/(V-sec) for Copper.
Table(1.10) Characteristics of Ge, Si and GaAs at 300 K.
Characteristics Symbol Units Ge Si GaAs Cu
Effective 1.04x10A
Density of Nc CmA -3 19 2.8x10419 4.7x104 17
States
Nv CmA -3 ee 1.02x10419 7x104 18
Energy Gap Eg eV 0.68 1.12 1.42
ie , 2.25104
Carrier ni CmA -3 1B 1.15x10410 1.6x104 6 8.5x10A22
concentration
mn
(unit
Effective mass mass 0.33 0.33 0.068
9.11x10
-31 Kg)
mp 0.31 0.56 0.56
(unit
mass
9.11x10
-31 Kg)
A ss
Mobility pn - 2(V- 3900 1350 8600 44
A ss
pn emia | aang 480 250
s)
Dielectric os 16.3 11.8 10.9
Constant
1 A
oe Cm -3 acest 5x10 22 4.42x10A22. —-8.5x10A22
Concentration 22
ao EBR Vicm 10\5 3x10A 5 3.5x10A 5
1.10.2.1. METALS ( with special reference to the mobility of conducting electrons and its implications for particle
accelerators ).
Metal is a lattice of positive ions held together by a gas of conducting electrons. The conducting electrons
belonging to the conduction band have their wave-functions spread through out the metallic lattice. The average
kinetic energy per electron is (3/5)Ep (This will be a tutorial exercise). Hence
2Z
1 _ie
ym ve
1.93
where m* is the effective mass of the electron but we will assume it to be the free space mass.
Therefore:
v.= Vi gE x =]
sinsesbsbinn seaess 1.94
Where
v, = velocity of the mobile electron in conduction band
This velocity is not thermal velocity but velocity resulting from Pauli’s Exclusion Principle which essentially is the
result of the ferm-ionic nature of electrons. Electrons tend to repel one another when confined in a small Cartesian
Space. Electrons are claustrophobic.
Therefore mean free path =
=VLXE
Where Tt is mean free time.
Substituting the appropriate values for each metal, we get the mean free path for electron in their respective metals.
Table(1.10) Tabulation of the Fermi Energy, velocity, mean free time and mean free path of conducting electrons in
their respective metals.
Metal EF Velocity(x104 5 m/s) t (femtosec) L*(A°)
Li 4.7 9.96 9 90
Na 3.1 8.08 31 250
K 2.1 6.65 44 293
Cu 7.0 12.15 27 328
Ag 5.5 10.77 Al 441.6
As we see from Table(1.10), the mean distance between two scatterers is 2 orders of magnitude greater than the
lattice constant which is of the order of 5 A°. Hence lattice centers per se are not the scatterers but infact the
disorderliness is what causes the scattering. The scatterers are thermal vibrations of lattice centers, the structural
defect in crystal growth and the substitutional/interstitial impurities. This implies that with reduction in
temperature mobile electrons will experience less scattering hence the metal will exhibit less resistivity leading to
positive temperature coefficient of resistance. We will dwell upon this in Section (1.12).
1.10.2.2. SEMICONDUCTORS
Semiconductors are insulators initially. At low temperatures, all electrons are strongly bonded to their host atoms.
Only at temperatures above Liquid Nitrogen that thermal generation of electron-hole pairs take place. So in
semiconductors the situation is quite different as compared to that in metal. The conducting electrons and holes
owe their mobility to thermal energy they possess in contrast to the conducting electrons in metal. On an average
by Equipartition Law of Energy, the mobile carriers possess (1/2)kT thermal energy per carrier per degree of
freedom. Since the carriers have 3 degrees of freedom hence they possess (3/2)kT average thermal energy per
carrier.
Therefore
actsianlovessbageencadvaslsissee sins cesssmsaeaeaperesessesecsmoscsesns 1.97
Therefore in semiconductors the mean free path will be the product of the thermal velocity and the mean free time.
Mean free time is calculated from mobility of the mobile carriers which is determined experimentally.
From Table(1.10) we obtain the mobility values. In Table (1.11) the mobility, mean free time, thermal velocity and
mean free paths are tabulated for Ge , Si and GaAs.
Table(1.11) Mobilities, Mean Free Times, Thermal Velocities and Mean Free Paths of Ge, Si and GaAs.
Semiconductor pin(cm/2/(V-sec)) t (femtosec) v e (m/sec) L* (A°)
Ge 3900 2217 0.95x104 5 2106
Si 1350 767.6 0.95x104 5 729
GaAs 8600 4890 0.95x104 5 4645.5
As we see electron has much larger mobility in semiconductors as compared to that in metals. This implies that the
mean free path of electrons is greater by one order of magnitude in semiconductor as compared to that in metal.
But why is the scattering less in semiconductors as compared to that in metal.? This answer is obtained by
determining the de Broglie wavelength of electron and by using wave optics.
We will determine the velocity of a conducting electron in Electron Microscope, in metal and in semiconductor. In
these three cases the conducting electron gains Kinetic Energy equal to the Potential Energy it loses while falling
through a potential difference of 10kV in case of Electron Microscope(because 10kV is the accelerating voltage in
Electron Microscope), through a potential difference of 4V in case of metal(because average kinetic energy
associated with conducting electron is (3/5)E; and Epis 7eV in copper) and through a potential difference 0.025V
in case of semiconductor ( since thermal voltage at 300K Room Temperature is kT/q= 0.025V). From the kinetic
velocity the de Broglie wavelength is determined. The set of equations are: Kinetic Energy gained =
r= q XV,
2xm°* acc
Therefore momentum gained
p= /2m*qV,
acc
5)
Therefore de Broglie wavelength:
h
A=t=
=- 5 oa
P v2m' Voce
In Table (1.12) the de Broglie wavelengths are tabulated:
Table1.12. de Broglie wavelengths of conducting electron in Electron Microscope, Metal and Semiconductor.
V acc ve (m/sec) A(m) Implications
ee 10kV 59x10 6 a m = (1/50) A<< a (lattice constant)
nooky — BB78*T0" x40 -12 m
Metal 4V 104 6 6x104 -10 m = (5A°) A~ a (lattice constant)
Semiconductor 0.025V 1045 TIS m= pos
(78A°)
constant)
As seen from Table(1.12), we see that de Borglie wavelength is much less than the lattice constant in case of
Electron Microscope. For 100kV , theoretically the resolution should be (1/100)(4A°) This is like Sunlight falling
through a broad aperture. Sun-ray will pass in a straight line and shadow of the aperture should fall on the screen
behind the aperture. Hence in an Electron Microscope, a regular lattice array does not scatter an electron beam.
The shadow of the crystal lattice should be imaged. But this theoretical resolution is never achieved since we are
using magnetostatic focusing. Only 1A° is the resolution actually achieved. In case of 10kV, though the theoretical
resolution (1/50)(5A°) but in practice only 10A° resolution is achieved. The electron beam can penetrate through a
thin specimen and produce the image of its broad features without being influenced by the atomic details.
In electron microscope the electron can be accelerated to higher energy to obtain a finer resolution. It can resolve
on the scale of molecules but can barely perceive the atoms.
To resolve at atomic and sub-atomic level we need to go to particle accelerators. Particle Accelerators are
gargantuan machines which can be regarded as giant microscopes for probing into the innermost recesses of
matter - an awesome complement to the giant telescopes which probe to the edges of the Universe .
To arrive at the resolving power of particle accelerator we must know Special Theory of Relativity and we must
make relativistic corrections in order to arrive at the correct resolving power of the particle accelerators. These are
described in the Appendix XX XXIV . Here we will just use them to arrive at the resolving power of the particle
accelerators.
Relativistic momentum is related to the total energy E by the following relation ship:
r= pe + mic*
aise dvionidodie ns sc siesicw slants sismeae sas siesieseeave ease ssissuseweseeeseeans 1.98
de Broglie wavelength associated with this particle is:
h_ he
P V[E?—-(2mge?) ]
sioceaigia.aig oes s'eg sla Seid doses Saleibelgl da d's ¥ssle Ssietois sp eicawle Sawasieveuseessls 1.99
Using Equation (1.99), the resolving power of various particle accelerators operational around the world is
tabulated in Table(1.13).[Taken from Table(9.1), “Overview of Particle Physics”, by Abdus Salam, New Physics,
edited by Paul Davies, Cambridge University Press, 1992]
Table 1.13. The resolution of the particle accelerators around the World.
Name & Location Energy reached Year Resolution Particle det:
Nucleussize
Alpha decay10MeV,alpha ‘ ;
article’s 4.5x104 1 aah
Rutherford* Manchester, UK. P Shades 1911 determined
velocity=2x107m/s, Alpha -15m
be 30fmBut
particle=4He nucleus; s nee
estimate is
1.24x104 Protonssize
a -15m 104-15m=1
1932 1.24104 Neutronssiz
-15m =1fm
1GeV 1.24x104
-15m
Bepc(et+e-)Beijing 4GeV 1987
TRISTAN(e+e-)Japan 60GeV 1987
1.24x104 ee
10GeV 1979 16m Quarkssize:
vol % 1.24x104
SLC(e+e-)Stanford, California, USA; 100GeV 1987 47m W-, W+ &
LEP(D) (e+e-)Large electron-positron 1.24x104
collidorCERN, Geneva; nee: oa -17m
6.2x104
LEP(II) (e+e-)CERN,Geneva 200GeV 1995 18m Top Quarks
A
HERA (ep)Hamburg 320GeV 1991, | o27se
-18m
A
SpSCERN, Geneva 900GeV tog, [yee
-18m
1.24x10A No excited
TevatronFermiLab,USA 1TeV 1987 quarks or le
-18m :
detectedsize
A
TevatronFermiLab,USA 2TeV 1987 Got
-19m
A
UNKSerpukhov, Russia 3TeV 1995 seca
-19m
A-
EeSerpukhov,Russia 4TeV ? tay 1
Large 16TeV ? 7.75104
HadronCollider(LHC),CERN, Geneva -20 m
SSC(super particle A0TeV ? 3.1x104
superconductingCollider), USA; -20m
1.24x104
i)
1PeV $ 21m
1.24x104
i)
1EeV $ 24m
* the first particle accelerator was established at Cavendish Laboratory, Cambridge University. In 1919 Rutherford
became the first Director and he was instrumental in establishing the particle accelerator.
In Metal the wavelength is comparable to the lattice constant. This is like light falling through a narrow aperture
whose dimension is comparable to the wavelength. Incident light will form a circular diffraction pattern behind the
aperture on the target screen. This implies that conducting electron in a metallic lattice is strongly scattered by the
lattice centers. Hence it has a very low mobility.
In Semiconductor, the de Broglie wave length is much larger than the lattice constant. Hence lattice scattering is
weak and only the gross imperfections cause the scattering. These gross imperfections could be phonons and
dislocations extending over several lattice constants. This is what makes conducting electrons much more mobile
in semiconductor as compared to that in metal.
In metal, conducting electrons behave like degenerate gas and not quite like ideal gas whereas in semiconductors
they behave like non-degenerate gas which is more like ideal gas obeying ideal gas law.
In ideal gas the molecules are far apart, independent of one another and possessing average energy of (3/2)kT
whereas in degenerate gas the molecules are closely packed and average kinetic energy is much larger than
(3/2)kT. In Table(1.14),
Metals and Semiconductors parameters have been tabulated in the same table.
Table(1.14). Conductivity(o), Fermi Level(E F ), Mean Free Path(L*) and Mean Free Time(t) at 0°C for
monovalent metals and semiconductors.
pecm2/(V- EF L*
Metal 0(106S/cm) p(Q-cm) n(10422/cm3) s)= o/(na) (eV) (A°) t(fs)=
Li 0.12 8.3x104-6 4.62 16.2 4.7 110 9
Na 0.23 4.35x10A-6 2.65 54.17 3.1 350 31
K 0.19 5.26*104-6 2.1 370 44
Cu 0.64 1.67x104-6 8.5 47 7.0 420 27
Ag 0.68 1.47x104-6 5.9 72 5.5 570 41
Ge 47 ni=2.25x10413 3900 2106 2217
Si 300k ni=1.15x10410 1350 729 767.6
GaAs 70.5M ni=2x1046 8600 4645.5 4890
Chapter 3. Special Classification of Semiconductors.Sec3.1.Compund
Semiconductors
Chapter 3 covers the special classification of Semiconductors. The first
classification is Compound Semiconductors . Section 3.1 is devoted to
Compound Semiconductors.
Chapter 3. Special Classification of Semiconductors.
Section 3.1. Compound - Semiconductors.
Compound Semiconductors are the basis of a whole new branch of Science
and Technology known as Photonics. Light Sources and Light Detectors
belong to this discipline. III-V elements give rise to Compound
Semiconductors which are suitable for Light Generation or Light detection.
These III-V elements form alloys across the whole range of concentration at
their growth temperature. This wide miscibility range allows alloys to be
grown with band structures adjusted for specific applications. This leads to
Band structure manipulation according to our specific needs. This is known
as Band-gap Engineering. The common Alloys used in Photonics are as
given below:
i. GaP(2.3eV, a = 5.42A°) GaAs,P(1-x) GaAs(1.42eV,
5.65A°): here x is the stoichometric coefficient and by adjusting ‘x’ ,
band-gap can be tailored from 1.42eV to 2eV.
ii. NP(1.3eV,5.85A°) InGaP GaP(2.3eV,
5.42A°).
ill. GaAs(1.42eV,
5.65A°) GaAlAs AlAs(2.2eV,
5.65A°).
iv. GaAs(1.42eV, 5.65A°) GaAsSb GaSb(0.65eV,
6.1A°).
v. GaAs(1.42eV, 5.65A°) GalnAs InAs(0.35eV, 6.05A°).
vi. InP(1.3eV,5.85A°) InPAs InAs(0.35eV,6.05A°).
vii. GaSb(0.7eV,6.1A°) GalInSb InSb(0.15eV,6.5A°).
The three element alloys are TERNARY ALLOYS. Two from Group II
and two from Group IV combine to form QUATERNARY ALLOYS.
Wide miscibility shown above has been translated into a topological
diagram Figure 3.1 where Band-Gap versus Lattice Constant is plotted for
the seven major Compound Semiconductors namely GaP, GaAs, GaSb,
InAs, InSb, InP , AlAs, and AlSb and their derived Ternary Alloys.
As seen from Figure 3.1, none of the pure compounds listed have a direct
band-gap more than 1.65um for producing Visible Spectra Radiation. GaAs,
InP, GaSb, InAs and InSb have direct band-gap but less than 1.65eV.
AIP, GaP, AlAs and AISb all have Band-gap larger than 1.65eV but all are
in-direct band-gap hence not suitable for optical generation. Hence for
Optical LEDs we go for alloys of GaP and GaAs known as ternary alloy
GaAsqj-x)Px. In Figure 3.2, the white light spectrum and the corresponding
Band-gaps are shown.
Figure
2.4
— Direct gap
--- Indirect gap
2.0
1.6
=
2
a 1.2
“he
‘7
3
Qa
Sh
Z 08
{aa}
0.4
5.4 5.5 5.6 D1 5.8 5.9 6.0 6.1 6.2 6.3 6.4 6.5
Lattice constant ag (Angstroms)
Bandgap energy and lattice constant of various III-V semiconductors at room
temperature (adopted from Tien, 1988).
www. LightEmittingDiodes.org
Figure 3.1.Topoloigical diagram for Compound Semiconductors and
their Ternary Compounds. The Solid lines indicate Direct Band Gap
materials and dashed lines show In-direct Band Gap materials.
Wavelength A (um)
P
2.36 eV —— AIP
2.33 eV , direct-indirect Ey= 2.45 eV
crossover (A = 532 nm) ay = 5.4510 A
3 (Al,Gay_.)o sIng sP
> x x). 54N0.5
oa (lattice-matched ~ GaP
uD to GaAs) = Eg =2.26 eV
Ss = ay=5.A512A
po R
5 % InP
a Sh n
¢ tn SB Eg=1.35eV
3 2 a) = 5.8686 A
= Direct -_ e
aus Gert GaA
--- Indirect gap GaAs : = —
[J (Al,Gay_,),Iny_,P | g
Jat al Gazln1-aP ap = 5.6533 A
5.3 5.4 3:5 5.6 5.7 5.8 5.9 6.0
Lattice constant a (Angstroms)
Bandgap energy and corresponding wavelength versus lattice constant of
(Al,Gaj_x)ylnj—P at 300 K. The dashed vertical line shows (Al,,Ga|_x)9.5Ing.5P lattice
matched to GaAs (adopted from Chen ef al., 1997).
www. LightEmittingDiodes.org
Figure 3.2. The plot of Band-Gap and the corresponding Wavelength
versus Lattice Constant.
Experimental external
quantum efficiency of undoped
and N-doped GaAsP versus the P
mole fraction. Also shown is the
calculated direct-gap (I) transi-
tion efficiency, Np, and the calcu-
lated nitrogen (N) related transi-
tion efficiency, Ny (solid lines).
Note that the nitrogen-related ef-
ficiency is higher than the direct-
gap efficiency in the indirect
bandgap (x > 50%) regime (after
Campbell ef al., 1974).
0.1
0.01
External quantum efficiency Next (%)
ie) 7 20 40 60 80 100
i x 0 www. LightEmittingDiodes.org
Phosphorus mole fraction x (%)
Figure 3.3. External Quantun Efficiency vs stoichiometric corefficient
of Phosphorous in ternary alloy GaAs (1-x) Px.
For the manufacture of coloured LEDS we have to use GaAsv1_,)P, ternary
alloy. This has a problem. Below x = 0.45 it is a direct band-gap material
but at higher proportion of Phosphorous it becomes in-direct and its
performance becomes very poor as shown in Figure 3.3. By doping with
Nitrogen it can be restored to Direct Band-gap material and utilized for
LED manufacturing. Table 3.1 tabulates the different ternary alloys used for
manufacturing the spectrum coloured LEDs.
Table 3.1.The Ternary Alloys (GaAs (1-x) P x ) used in the manufacture
of the whole range of spectrum coloured LED. Here E g
=1.424+1.15x+0.176x 2.
Energy
Colour Wavelength(pm) seine ‘x’ Substrate
Red 0.64 1.9 0.4 GaAs
Orange 0.62 2 0.5 GaP
Yellow 0.58 215 0.58 GaP
Green 0.55 22D 0.656 GaP
Blue 0.475 2.60 0.9 SiC
3.1.1. The Crystal Structure of Compound Semiconductors and its
Dopent.
GaP, GaAs, GaSb, InAs, InSb, InP , AlAs, and AlSb have the the same
crystalline structure as Diamond but now it is called Zinc-Blende. It is two
interpenetrating FCC sublattices with one sublattice made of Group III
element and the other FCC sublattice made of Group V element and one
sub-lattice is displaced with respect to the other along the diagonal of the
cube by a quarter of the diagonal length i.e. by aV3/4. The net result is that
every Group III has FOUR Group V atoms as neighbours and similarly
Group V has for Group III atoms as neighbours. This completes the co-
valent bond requirement.
Table 3.2 tabulates the dopents of II-V Compound semiconductors.
Table 3.2. P and N type dopents for III-V Compunds.
Group II Ill IV Vv VI
Zn Ga Si As S
In Ge P Se
N
Acceptor Ampuoteric Donors
Dopents
Silicon and Germanium can either be Donor or Acceptor depending upon
what they substitute. If Group III is substitute then they become DONOR
and if they substitute Group V then they become acceptor. But since Si is
smaller in size hence energetically it is favourable to replace Ga hence Si is
Donor in GaAs and bigger Ge substitutes the bigger As atoms hence Ge is
acceptor.
3.1.2. Band-gap Engineering.
The manipulation of Band Structure required for different kinds of
applications is Band-gap Engineering. There are three techniques of Band-
gap Enginering:
i. Alloying;
ii. Use of Heterojunctions and
iii. Built-in strain via mismatched epitaxy.
The Aim of Band-gap Engineering is to tailor/customize the band-gap
according to the wavelength at which we want to operate.
The second objective is to tailor the lattice constant according to our
matching or our mis-matching requirements.
The wide miscibility range allows alloys to be grown with Band structures
adjusted and finally tuned for specific applications.
3.1.3. Properties of Alloys.
In Alloys we have Lattice Parameter (a)Law called Vegard’s Law. If we
have two solid mixture AxBj-x) then the Alloy’s Lattice Parameter is given
as follows:
Aauoy = X-A, + (1—x).ag a4.
Alloys are not perfect crystals even if they have perfect lattice structure.
This because in solid mixture atoms donot have periodic placement.
By virtual crystal approximation:
Eo"? = x. EA + (1—x). EF 3.2
We have quadratic approximation also:
Ege? =a+b.x+c.x? a3,
Equation 3.3 is the same as the Equation given in Table 3.1.
Alloying induced dis-order causes a BOWING Parameter in compound
semiconductor Wafer. Equations 3.2 and 3.3 are valid only if the alloy is a
good mixture i.e. perfectly random mixing.
In an alloy AxBy-x) a good mixing results into the fact that the probability
that A is surrounded by B is (1-X) and B is surrounded by A is X. If
proportion is different from the stichiometric corfficient then it is clustered
or phase repeated.
Test of Eq (3.2):
AIAs has Eg = 2.75eV and GaAs has E, = 1.43eV therefore in Alp 3Gao,.7As
has
Eg = 0.3x2.75+ 0.7x1.43 = 1.826eV.
Chapter 4.Light and Matter — Dielectric Behaviour of Matter
Chapter 4 gives the historical background of Optics and its applications to
matter.
Chapter 4.Light and Matter — Dielectric Behaviour of Matter
[“OPTICS” by Eugene Hecht, [IIrd Edition, Addison-Wesley-Longman
Incorporation, Readings, Massachsetts].
Section 4.1. Brief History of Scientific and Technological Development
in the field of Light.
Study of Light and its application in Human Society can be traced back to
1200 BCE. In Exodus 38.8 (a chapter in The Bible) we find the mention of
“Looking Glass of the Women”. Early mirrors were made of polished
copper and bronze. Evidence of the use of mirrors turned up in excavations
of the workers’ quarters near the construction site of Pyramid during the
ancient Egyptian Civilization. At that time it was made of Speculum —a
copper alloy rich in tin. In Roman Civilization we find the use of convex
lens as Magnifying Glass as well as Burning Glass.
After 475CE, with the fall of Western Roman Empire, dark ages descended
in Europe and the center of Scientific Enquiry shifted to the Arab World.
Islamic scholar Abu Ali al-Hassan Ibn al-Haytham (CE. 965-1041), known
in the West as Alhazen, began his career as just another Islamic polymath.
He was put in House Arrest by Al-Hakim, the Calipha of Cairo, because he
failed to regulate the flow of Nile river. While in House Arrest for 10 years,
Alhazen revolutionized the study of optics and laid the foundation for the
scientific method. (Move over, Sir Isaac Newton.) Before Alhazen, vision
and light were questions of philosophy. Alhazen considered vision and light
in terms of mathematics, physics, physiology, and even psychology. In his
Book of Optics, he discussed the nature of light and color. He accurately
described the mechanism of sight and the anatomy of the eye. He was
concerned with reflection and refraction. He experimented with mirrors and
lenses. He discovered that rainbows are caused by refraction and calculated
the height of earth’s atmosphere. In his spare time, he built the first camera
obscura.
From 1000 CE to 1600 CE, there was only a modest revival of Scientific
Enquiry and Research in Europe. The science of vision error correction by
the use of Eye Glasses were introduced. Looking Glass or Mirror
Technology was revived by the use of liquid amalagam of tin and mercury
coated on the back of glasses. Use of multiple mirros and use of positive
and negative lenses were in vogue in this era.
Section 4.1.1. The revival of Scientific Enquiry in 17 th Century in
Europe -The Renessaince Period.
Table 1. Time-line of Scientific Research in Application of Light for
Human Good.
Time Scientist Subject
Hans
Lippersley
October (1587- Applies for a patent for the
2,1608 1619)Dutch Refracting Telescope.
Spectacle
Maker
Galileo Galilei
1609 (1564-1642),
Padua, Italy.
Builds his first Telescope for
Astronomical Observation.
Astronomical Four satellites around Jupiter which
1609 Discoveries of are lo, Europa, Callisto and
onward Galileo Ganymede. Galilie called them
Galilei. Medicean Stars;
Time
After
1609
After
1609
1657
Scientist
Zacharius
Jansenn(1588-
1632),
Dutchman.
Willebrored
Snell (1591-
1626)
Pierre de
Fermat (1601-
1665)
Subject
Observed and analyzed the Sun
spots;
He discovered the poke marked
surface of Moon concluding that it
had mountains and craters;
He saw the phases of Venus which
lent support to the Copernican
World View;
He identified Milky Way as our
Home Galaxy. Till then it was
mistakenly regarded as nebular
cloud;
He observed Neptune but could not
identify it as one of our Planets;
Invented Compound Microscope.
Discovered the Law of Refraction
and measured Refractive Index of
many mediums.
Rederived the Law of Reflection
using the Principle of the Least
Time.
Time Scientist Subject
Interference and diffraction was
observed.
But there was no consensus about
the nature of light-is it a Wave or
is it Corpuscular ?
Within a year of death of Galileo, Sir Issac Newton(1642-1727) was born in
England. He studied the dispersion of light into seven rainbow colours
(Violet, Indigo, Blue, Green, Yellow, Orange and Red) but he was unable to
reconcile the rectilinear propagation of light with the spherical wave-front
of light from a point source. Hence Newton favoured Corpuscular Theory
of Light.
There was a problem of Chromatic Aberration in Refracting Telescope
which Newton was unable to correct. So in its place he built Reflecting
Telescope in 1668. This was only 6 inches long and 1 inch in diameter but it
had a magnification of 30 timesa.
Christian Huygens (1629 — 1695) extended the Wave Theory of Light and
gave the theoretical basis of Reflection, Refraction and Double Refraction.
He discovered that there was perpendicular polarized light (perpendicular to
the plane of incidence) and parallel polarized light (parallel to the plane of
incidence).
Ole Christensen Romer (1644-1710) was the first person to recognize that
light was not instantaneous but it had a finite velocity. Based on this
reasoning in 1676, Romer predicted that on 9th November Io, a moon of
Jupiter, would emerge from Jupiter’s shadow 10 minutes later than what
would be expected from the yearly average motion. This what happened
and this led to the conclusion that light took 22 minutes to cover a distance
equal the orbital diameter of Earth around Sun i.e. a distance of 2
Astronomical Unit (A.U.) where 1 A.U. = 1.49598x10!'m. Using this data
we airive At c = 2.2664x10°m/s but the correct value is c = 3x10°m/s. This
error occurred due to underestimation in Jupiter’s Orbit size.
Huygen and Newton based on the same reasoning arrived at c = 2.3x10°m/s
and c = 2.4x10°m/s respectively.
Section 4.1.2. 19 th Century — the emergence of Wave Theory of Light.
Dr. Thomas Young (1773-1829) in England presented a series of papers on
Wave Theory of Light in 1801, 1802 and 1803 before the Royal Society of
London. He added a new dimension to the existing Wave theory by
illustrating the Principle of Interference. He explained the coloured fringes
in thin films and determined the wave-length of the seven colours of light.
Augustine Jean Fresnel (1788-1827) in France independently explained the
diffraction pattern arising from various obstacles and apertures and he also
accounted the rectilinear propagation of Light based on Wave Theory.
In 1725, James Bradley (1693-1762) attempted to measure the distance to
star by triangulation method by observing a given star at six months time
period. During this experiment he observed Stellar Aberration. This is
different from Parallex Error.
The problem of perpendicular and parallel polarization led Young to revise
the mode of propagation of Light. Initially it was assumed to be
longitudinal much as the sound waves. But detection of polarization forced
Young to postulate that Light had Transverse Mode of propagation.
By 1825, Wave Theory of Light was established.
In 1849, Armand Hippolyte Louis Fizeau (1819-1896) did the first
terrestrial determination of the speed of light in air. It came to be 315,300
km/s.
Subsequently Focault measured the velocity of light in water. It turned out
to be less than ‘c’. This was interoperated as some kind of drag effect by the
water medium. This also contradicted Newton’s Corpuscular Theory of
Light.
In 1845, Michael Faraday (1791-1867) established that a strong magnetic
field could change the Polarization of Light Beam.
In 1861 and 1862, James Clerk Maxwell (1831-1879) synthesized the
empirical knowledge of Gauss’s Divergence Theorem, Faraday Induction
Law and Ampere’s Circuital Law into four differential equations and he
established that Light was Electro-Magnetic Field propagating in all
pervading aether as a transverse wave at a velocity c in vacuum:
1
¥V Hof
c= where t, = absolute permeability and ¢, = absolute permittivity 1
In a dielectric medium the velocity of propagation (v) is:
1 1 c c
v =—__ [na non— magnetic Dielectric = ———_ = —= - la
V Hr Hof, £o V Hy Hoe,E VE, ”
Where n = refractive index of the material.
From (1a) it is evident that n = refractive index = Ve, .
From Maxwell equations it was clear that Light was an Electro-Magnetic
disturbance propagating out of a point source with the velocity of light with
a spherical wave-front.
The Four Maxwell Equations in differential form are:
V.D=p or V.E= .
££,
= dielectric constant. This is Gauss'sLaw which states that total
where D= £,£,E Here «, = relative permittivity
electric flux coming out of a given volume is equaL
to the total electric charge enclosed by the volume. 2
6B
VxE=- ro this directly followsw from Faraday's Induction Law
which states that total induced emf in a Copper Wire Coil is equal
to the rate of change of magnetic flux cutting the copper coil. 3
V xX H =],(conduction current density) + = (displacement current)
this directly follows from Ampere Circuital Law which states that the line integral
of Magnetic Field H is equal to the current enclosed by the integral path 4
V.H =0 this follows from the fact that there is no Magnetic Monopole 5
From (2), (3), (4) and (5) it is evident that:
i. There is a general perpendicularity of E and H;
ii. The Maxwell Equations are symmetrical;
iii. E and H are interdependent.
Time varying E field through (4) produces H field which is perpendicular to
the direction of change of E and time varying H field through (3) produces
E field which is perpendicular to the direction of change of H. So as an
Electro-Magnetic disturbance is produced self sustaining transverse Electro-
Magnetic Field travels out from the source of disturbance as shown in
Figure 4.1. E and H are coupled in form of a pulse. E generates H farther
out and H generates E still farther out. Thus E-M wave travels out in self
sustained fashion and there is no need of aether. Still Maxwell assumed that
there was an all pervading luminiferous aether which helped the
propagation of E-M waves.
—>-
The electric charge is
accelerated in positive X-
direction.
Figure 4.1.A Kink in the E-Field due to acceleration of the charge Q
If a charge is accelerated in positive X-direction as shown in Figure 4.1.
then a kink is produced in the Electric Field at (c.dt). This kink travels out
at ‘c’. It has a radial as well as transverse component. Radial
Component(electro-static component) diminishes as the square of r while
Transverse Component(radiative component) diminishes as ‘r’. Hence after
some distance r, only Transverse Component remains which detaches from
the electric dipole and travels out as self-supporting E-M Radiative Field.
In 1888, Heinrich Rudolf Hertz(1857-1894) generated long E-M waves and
published the results in the Philosophical Transactions of Royal Society of
London.
In 1881, Michelson-Morley completed the measurement of velocity of light
in different frames of reference but could detect no change in the standard
value of ‘c’. Light velocity seemed to be invariant of the source of emission
velocity.
Section 4.1.3. The Conceptual Paradigm Shift at the turn of 19 th and
20 th Century.
The invariance of the velocity of light to the Frame of Reference
necessitated a paradigm shift in the theoretical framework given by
Newtonian Relativity. Jules Henri Poicare(1854-1912) was the first to grasp
this invariance and he expressed an alternative view point:
“Our aether — does it really exist ? I donot believe that more precise
observations could ever reveal more than relative displacement.”
In 1905, Albert Einstein introduced his special theory of Relativity in which
he showed that,
“the introduction of ‘Luminiferous Aether’ will prove superfluous in as
much as the view here to be developed will not require an absolutely
stationary space.” E-M wave was now envisaged as a self supporting and
self-sustaining process without the need of a sub-stratum.
4.1.3.1. Special Theory of Relativity.
Newtonian Mechanics fails the test of invariance of the speed of light from
frame to frame. Einstein resolved this failure of Newtonian Mechanics.
Time measured by a clock in a given frame of reference is called PROPER
TIME.
In moving frame, time measured by a clock placed in the same frame of
reference is:
z
ds=c.dt' or dt'===dt az
Therefore
dt(Rest Frame) = dt (Moving Frams)
ot
1 ——-
= Time Dilation 6
Equation (6) implies that time slows down in moving frame as observed
from rest frame. That is a person in a moving frame will be seen to age
slowly when observed from the rest frame
Some examples of Time Dilation:
Example 1. This experiment was carried out in CERN.
Positive Kaons have a rest lifetime of tx = 0.1273us.
In the lab, it is generated at a velocity of 0.927c.
At this Relativistic Speed its observed life-time = 0.879357 us.
This is the result of Time Dilation.
Distance travelled in the Lab before it decays is = 0.927cx0.879357Us =
244m
Example 2.
Cosmic Rays are continuously bombarding our atmosphere. Cosmic Rays
consist of PROTONS.
Protons collide with Nitrogen Molecule to form to form Pions. Pions decay
to Muons. This occurs at a height of 45Km. Muons belong to Leptons.
Electron and Tau-lepton belong to the same group of the particles.
Velocity of Muons = 99.9943% of c.
Its rest lifetime = 1,9 = 2.2us. That is in rest frame the lifetime is 2.2ps.
If it was travel at 99.9943% of c for its rest lifetime of 2.2us it would travel
only 659.5m and it could never be detected on the ground detectors.
But it is detected.
This is because its observed lifetime from Ground is 0.0192988s and it is
travelling at 99.9943% of c. Hence distance travelled as observed from the
ground = 5785.3Km. Muon has been generated 45Km above the Ground.
Hence it easily reaches our Ground and is readily detected.
l(length measured in moving frame)
| 2
= 1,(proper length of a body in Rest Frame) |1— —
c
= Contraction of length 7
In a moving frame transverse dimensions remain unchanged.
Let Vx , Vy , Vz be the velocities of a body in rest frame.
Let vx , Vy , V, be the velocities of a body in a moving frame. The moving
Frame is moving with velocity V in (+)ve x-direction.
Transformation of Velocities from Moving Frame to Rest Frame is given as
follows:
If the observation is instantaneous then velocity of light ‘c’ — infinity and
(8) reduces to:
v, =v, +V; %, =v, ;
v,=v, therefore v.= v,— V; 9
While sitting in a moving train, moving in (+)ve X direction with velocity
V, if we measure the velocity of Wind which in rest frame is moving in
(+)ve X direction at a velocity v, then in moving frame the observed or
measured value of the velocity of Wind is:
v, =v, —V; vy = v3, = 0,3 10
But velocity of light is not infinity hence measured value of the velocity of
Wind from the moving frame is given by equation (8).
4.3.1.2.Michelson-Morley Experiment and the invariance of the velocity
of light.
Velocity of PHOTON from a moving train is given by Equation (8).
2 +V
Therefore v, = ——; 11
14%
c
Rearranging the terms we get:
vv :
v(1+ r)=s+y 12
c
Open the bracket in (12);
UxVy ,
@ —_ ) =7,t+V 13
¢c
Put dashed terms on LHS and undashed terms on RHS:
' v,v,V —— V ' (1 =) —_ V th f _ (v, = V) 14
v, 2 Ue orv, 2) (v, —V) erefore v= (1 BY)
c
But in Rest Frame v, = c for a photon therefore measured in a moving
frame:
a — ee — 15
The set of transformations achieved by Equation (8) has to be postulated in
order to arrive at the invariance of velocity of light as demonstrated by
Michelson-Morley.
Section 4.2. Dielectric and its Physics.
Section 4.2 gives the physical basis of chromatic dispersion of light by a
prism.
Section 4.2. Dielectric and its Physics.
Non-conducting materials are defined as dielectric. The surrounding sea of
air is dielectric. Lenses, prisms and films are dielectric. Once light enters a
dielectric we must consider [! =—o X [, and € = € X €,_ That is relative
permeability and relative permittivity must be accounted which in vacuum
are unity.
Dielectric which are transparent in visible range are non-magnetic. Hence
by the definition of Refractive Index n = c/v = Ve,. This is Equation 1a in
Section 4.1.
What does TRANSPARENCY mean in materials science ? Light can pass
through a transparent material without absorption. This means light does
not interact with transparent material. In scientific language it means that
light does not interact with the medium it is passing through. If it interacts
then the medium is not transparent.
Rewriting Eq 1a we get:
n= Je, = Kz 6
In Equation 5, Refractive Index ‘n’ is measured using visible light and the
argument of the square root is Static Dielectric Constant which is quite
different from High Frequency Dielectric Constant because of the basic
physics which causes relative permittivity. Let us examine Water. The
Refractive Index of Water is 1.333 but static dielectric constant of water is
80. So it is evident that refractive index is frequency dependent. This was
evident some 300 years ago when Newton managed to disperse the Sun-
light into seven colours of a Rain-bow. The fact that we can disperse light
means that Refractive Index is frequency dependent.
4.2.1. Why is Refractive Index Frequency Dependent ?
In vacuum:
D = Electric Flux Density = £)Eyac 7
In a dielectric material:
D = £9£,Eugp 8
In (7) and (8), D are the same and relative permittivity of Dielectric is
greater than Unity hence the same Electric Flux creates a lower Electric
Field in the medium as compared to that in the vacuum. So how is the
remaining flux density accounted for. The remaining Flux density is causing
Polarization and the scenario is the following:
D=P+ £9Eysp = £0£,Euep = Eugen therefore P = (€— &))Eysp 9
The Application of an Electric Flux Density D in a dielectric medium
causes an Electric Field as well as it induces Polarization.
In polar medium such as water, the dipoles get aligned along the line of
Electric Field. This is defined as Orientational Polarization = Porientational:
In non-polar medium, applied flux density distorts the electron cloud with
respect to the nucleus of the atom as a result center of electron gets
displaced with respect to the center of nucleus and a dipole is created
temporarily. This is called Electronic Polarization = Pelectronic:
In ionic crystals such as NaCl, the application of Electric Flux displaces
positive and negative ions inducing dipole moments and producing Atomic
Polarization or Ionic Polarization = Patomic-
In Table 4.2. we give some examples of polar and non polar dielectric
gases.
Table 4.2. Assorted Molecules and their dipole moments.
Net dipole
Molecules Configuration moment
CO, tve and —ve charge centers Fac
coincident
H,O +ve and —ve charge displaced 6.2x103° C-m
HCl tve and —ve charge slight 0.401029 C-m
displaced
CO +ve and —ve charge displaced 3.43x10°9 C-m
At low frequency or under DC condition:
r= Porient © P stectronic . Pasomic = £o(¢, = 1)Eyep 10
Due to the contribution of all these factors to polarization depending upon
the situation, Static Dielectric Constant is very high. In water it is as high as
80.
But as the sinusoidally varying field’s frequency is increased because of
appreciable moments of inertia, polar molecules are unable to keep up with
the alternating field and their effective contribution to the net polarization
decreases. In Water dielectric constant is 80 up to 10!°Hz but it rapidly falls
off beyond that frequency. At Peta Hz(in visible part of spectrum) it falls of
to. 1,78:
In electronic polarization, electron cloud has no problem in following the
dictates of the harmonically varying electric field. Hence electronic
polarization makes its contribution right up to the Peta Hz.
But electronic polarization gives rise to a resonance absorption phenomena.
An atom can be treated as an harmonic oscillator with a central restoring
force F = -kx where k = spring constant or restoring force constant and x is
the incremental dispklcament from its equilibrium position. The harmonic
oscillator appears as shown in Figure 4.2..
electron cloud
electron cloud
Figure 4.2.The mechanical osci odel for an atom surrounded by a
spherically symmetric electron cloud.The oscillator can vibrate equally in all
directions.
The equation of motion of this atom with a spherically symmetric electron
cloud surrounding the nucleus is as follows:
Perturbing force = F, = eE(t) = electron cloud is subjected to an electric force
The equatiojn of motion of this harmonic oscillator is:
eee we ee I 11
eE, cos(w x= Mes
The first term on L.H.S. is the electric perturbing force. The second term is
the restoring force where k is the spring constant. R.H.S. D’ Alembert’s
force due to acceleration of perturbation of electron. (11) is a second order
linear differential equation with a Complementary Function + Particular
Integral as its total solution.
Under forcing function its harmonic oscillation is as follows:
e
k
m
= — 2 her = j— 12
x(t) OF 3 E(t) where wp | ;
(12) implies that whenever the incident light has a frequency equal the
natural frequency wg resonance occurs and the incident light is completely
absorbed by the medium as dissipative absorption.
From (9):
e= e+ E but P = exN = density of dipole moments and N
= contributing electrons per unit volume 13
Substituing (12) in (13) we obtain:
e
exN
ie a =
@-ay™
= £ +
But we know from (1a) that refractive n?= €/e9. Therefore Dispersion
Relation where Refractive Index is expressed as a function of frequency is
as follows:
Ne? 1
t= 1+ : ( 3 :) 15
4.2.2. Two cases of Refractive Index — below and above Resonance
Frequency (@ 0 ).
Case 1: ® < @g, P and applied E are in phase and n(@) > 1. This kind of
behaviour is generally observed in the real World around.
Case 2: @ > @p , P and applied E are 180°
out of phase and n(q@) < 1.
The plot of Equation 15 is given in Figure 4.3.
Absorption Band
Infrared fo1 Visible {92 Ultraviolet {093 X-Ray
Figure 4.3. Refractive Index versus Frequency.
If the incident photon (h*frequency)is not strong enough to excite the
crystalline atom then the incident photon is scattered or redirected. This is
called Ground State non-resonant scattering.
If the incident photon is equal to excitation energy then the atom will be
excited from ground state to one of the permissible higher energy
states.Subsequently atom will relax to the ground state and release its
excess energy as thermal energy. This is known as dissipative absorption.
If incident photon is less than the excitation energy quanta but matches the
natural frequency of the electron cloud system around each atomic nucleus
then the alternating Electric Field of the incident photon will set the
electron clouds in to oscillation. The crystalline atom continues to be in
Ground State but electron cloud in each atom is set into a weak oscillation.
This oscillatory vibration has two consequences:
1. Displacement of the center of the electron cloud with respect to the
center of the nucleus giving rise to dipole moment qd where q = electronic
charge and d = displacement of the center of the negative charge with
respect to the center of the positive charge.
2. Due to alternating electric field the dipole moment is also oscillatory.
Oscillatory dipole is accompanied with dipole elecrto-magnetic radiation in
accordance with the classical electro-magnetic Maxwell Laws. The
frequency of the secondary radiation is the same as the frequency of the
incident radiation. Since electron cloud plus the nucleus form a perfectly
elastic system hence coefficient of restitution is Unity. Just as a perfectly
elastic steel ball will continue to bounce up to eternity because it is a
perfectly conservative system. In exactly the same way the incident photon
sets the oscillatory dipole in motion and this oscillatory dipole re-radiates
the same energy at the same frequency but with an altered direction.If the
incident light is unpolarized then reradiation takes place in random
direction.This is known as elastic scattering.
During irradiation, the setting up of dipole oscillators and secondary re-
radiation is rapid. If the emission life-time is of the order of 10/-8 sec then
dipole oscillator will re-radiate 1048 photons per second.
If incident frequency is equal to the natural frequency of the electron cloud
system then amplitude of displacement is large leading to a strong dipole
moment and causing a large cross-section of absorption. At a modest value
of 10A2W/m/2 incidence 100million photons will be re-radiated.
When a material with no resonance in visible spectrum is bathed in White
Light, non-resonant scattering takes place and this gives each participating
atom the appearance of being a tiny source of spherical wavelets. As the
incident frequency becomes closer to the resonance frequency of the
electron cloud system the more strongly will the incident photon interact
with the dielectric material. In dense medium the strong interaction will
lead to dissipative absorption.
It is the selective absorption at resonance frequencies that creates the visual
appearance of the multi-colour surrounding- colour of our hair, skin,
clothes, colour of leaves, apples and paint.
In real life there are three absorption bands for various dielectric and lie
across a broad band of spectrum extending from Infra-red to Visible to
Ultra-violet to X-Ray. Below IR these absorption bands can lie in RF band
of spectrum also.
Below the absorption band lies the Normal Dispersion Band of frequencies.
In this band of Spectrum there is a positive slope of the refractive Index
curve as shown in Figure 4.3. Higher frequencies have higher Refractive
Index. Therefore in chromatic dispersion of white light by a glass prism,
Red Light bends the least and Violet bends the most.Here there is elastic
scattering of light. Light is absorbed and re-radiated without any
dissipation. Hence the term Elastic Scattering.
In the absorption band, energy of the incident light is absorbed and
dissipated in form of heat. This is called Dissipative Absorption. In
absorption band the refractive index has a negative slope as shown in
Figure 4.3.
After the absorption band , n < 1 (as seen in Figure 4.3) meaning by light
has a phase velocity greater than light. But the group velocity, the velocity
at which energy travels, remains below velocity of light.
It is evident from the discussion that over the visible region of the spectrum,
electronic polarization is the operative mechanism determining the
refractive index dependence on frequency. When the incident frequency is
lower than the resonance frequency/characteristic frequency/natural
frequency then oscillations amplitude is very small and incident light is
elastically scattered. There is no dissipation. At resonance the amplitude is
increased and dissipative absorption sets in. The material becomes opaque
in dissipative absorption bands though it remains transparent at other
frequencies.
Section 4.3. Dielectric Loss.
Section 4.3 describes the Dielectric Loss and Loss Angle.Section 4.4 describes the
piezoelectric material and Quartz CrystaL Oscillator.
Section 4.3. Dielectric Loss.
Theoretically the dielectric constant is real and a capacitor having a dielectric
separation of plates always causes a 90° Leading Current with respect to the
applied voltage hence loss is zero and an ideal capacitor is always a conservative
system. But as we have seen that dissipative absorption can take place at higher
frequencies. The relative permittivity at alternating frequency is lower than DC
relative permittivity and relative permittivity becomes complex at frequencies
where loss occurs.
Thermal agitation tries to randomize the dipole orientations whereas the applied
alternating field tries to align the dipole moment along the alternating field. In
process of this alignment there is inevitable loss of electric energy. This loss is
known as Dielectric Loss. The absorption of electrical energy by a dielectric
material subjected to alternating Electric Field is termed as Dielectric Loss.
Under DC condition «, is real but at high frequencies
relative permittivity ¢, = ¢). — je,’ 16
The real part is the Relative Permittivity and the imaginary part is the Energy Loss
part. Because of Complex Relative Permittivity a loss angle (6) is introduced.
4.3.1. Loss Angle (8).
Parallel plate capacitor is given as follows:
__ &,€9A
- d
= distance between the plates 17
where A= cross — sectional areaof the Capacitor and d
To account for the lossy nature of the dielectric we assume complex relative
permittivity. Hence we get:
(el — jel")eqA ely
: £)E,A
c = ———— = ¢,—jC, where C,= and C, = a 18
Real part of the Capacitance causes Quadrature Component and Imaginary Part
causes In-Phase component. The In-phase component causes the loss angle hence
loss angle is defined as:
eo
Tan(d) = —
FS
3
19
In the Table 4.3.1.we tabulate some important dielectrics and their loss Tangents.
In Figure 4.4 the Relative Permittivity Real Part and Imaginary Part is plotted as
frequency.
Table 4.3.1. Some important dielectrics and their loss angle tangent.
Ceramics
Air
AlbOs
SiO»
BaTiO;
Mica
Polystrene
Polypropylene
SF. Gas
Tan (5)
0.002
to 0.01
0.00038
0.0001
to 0.02
0.0016
0.0001
0.0002
Dielectric
Strength
31.7kV/cm at 60
Hz
10MV/cm at
DC
79.3kV/cm at 60
Hz
Applications
Tested in 1cm gap
IC Technology
MOSFET
Low loss Capacitance
Low loss Capacitance
Used in High Voltage
Circuit BreaakersTo
avoid discharge
Ceramics Tan (5)
Polybutane
Transformar
Oil
Borosilacate
Glass
ga tidlinios and
Space charge
‘Low Siok Radio Wave
Dielectric
Strength Applications
>138kV/cm at Liquid dielectric in
60 Hz cable filler
128kV/cm at 60
Hz
10MV/cm
duration
10pus6M V/cm
duration 30s
Orientational Dipole
Electronic
Polarization
Ionic Polarization
€
=1
106 Hz 10°12Hz 10“°16Hz
Infra-Red Ultra-Violet
IN-D L\___-=
Figure 4.4. Plot of Imaginary Part and Real Part of Relative Permittivity.
As seen in Figure 4.4, there is significant loss at low frequency, at Radio-Wave
frequency, at Infra-Red frequency and at Ultra-Violet frequency. These correspond
to the natural frequencies of the electron cloud system shown in Figure 4.3. For
High-Q systems we require capacitance with dielectric material having a very low
loss angle. These are generally Poly-sterene Capacitances or Poly-propylene
Capacitors.
Section 4.4. Piezoelectric Effect and Piezoelectric Materials.
Electricity resulting from Pressure is known as piezo-electricity. This is called
piezo-electric effect.
Electricity causes deformation of such materials. This is known as inverse piezo-
electric effect. The most commonly used piezo-electric materials are Quartz,
Rochelle Salts, Sodium Potassium Tartarate and tourmaline.
Rochelle Salts are mechanically weak but electrically very sensitive. Hence used in
micro-phones, heads-phones and loud speakers.
Tourmaline are mechanically the strongest but electrically least sensitive. At
frequencies higher than 100MHz, vibrational breakage can take place hence
mechanically strongest materials are used namely Tourmaline.
Quartz Wafers are very popular as the stab lest electronic oscillators. These are
known as Quartz Crystal Oscillators and to date these are stab lest with only 1part
in million drift due to temperature, aging or load. Recently MEMS oscillators have
proved to be even more stable. In Quartz Crystal Oscillators, Quartz mechanically
oscillates but because of its piezo-electric property it behaves like a LC Tank-
circuit with a very high Q Factor. Hence it allows the electronic oscillator to
oscillate at its Resonance Frequencies which are critically dependent on the
Physical Dimensions. Hence as long as Physical Dimensions are accurately
reproduced so long the requisite Oscillation Frequency is accurately generated.
The resonance frequencies of some of the standard cut Quartz Wafers are given in
Table 4.4.1.
Table 4.4.1. Resonance frequencies and the Q-Factor of standard cut Quartz
Wafers.
Frequecy(Hz) 32k 280k 525k 2M 10M
Cut a DT DT AT AT
Bar
Rg(Q) 40k 1820 1400 82 5
Ls(H) 4800 25.9 12.7 0.52 12mH
C,(pF) 0.0491 0.0126 0.00724 0.0122 0.0145
Frequecy(Hz) 32k 280k 525k 2M 10M
C,(PF) 2.85 5.62 3.44 4.27 4.35
Q Factor 25,000 25,000 30,000 80,000 150,000
The electrical analog of the mechanical vibration of Quartz Crystal is as follows:
Electrical Analog of the Mass of the Quartz Wafer is Ls.
Electrical Analog of the spring constant of the Quartz Wafer is Cs.
Electrical Analog of the damping of the Quartz Wafer is Rs.
Cp is the parallel electrode capacitance.
Ls , Cs , Rs comprises the intrinsic series resonance path and Cp is in parallel with
this Series Resonance Path as shown in Figure 4.5.
Quartz Crystal Equivalent Model
Quartz Crystal Equivalent Circuit
Metallised
zw Electrodes
Crystal
=
ia =
Represents
Inertia, Friction © Represents self
and Stiffness Capacitance
of Crystal of Crystal
Figure 4.5. Quartz Crystal Electronic Equivalent Circuit
Quartz Crystal Reactance
i fe
Crystal Reactance
i]
Capacitive
Reactance
—— it
I
jx Fundamental
Frequency
Figure 4.6.Quartz Crystal Reactance Plot and Zero-Pole Pattern.
Section 4.5. Piezo-electricity and Ferro-electricity.
Section 4.5. describes ferro-electric materials and its characteristic
parameters.
Section 4.5. Piezo-electricity and Ferro-electricity.
Piezo-electric effect was discovered by Jacques and Pierri Curie in 1880.
They discovered that certain materials like quartz, Rochelle Salt, tourmaline
and Sodium Potassium Tartarate exhibited polarization on the application of
mechanical stress as shown in Figure 4.6.
No strain - ho potartzation Compressive Strain causes
negative polarization.
Elongation Strain causes
positive polarization.
Figure 4.6. Response of Piezo-electric material to Elongation and Compresive Strain.
Application of electrical field causes mechanical deformation. This is called
Inverse Piezo-electric Effect. Using Inverse Piezo-electric Effect, Ultra-
sonic transducers can be built. Alternating Electric field applied in the
frequency range 20kHz to 100MHz will set the piezo-crystal in mechanical
vibration at the same frequency as long as the frequency happens to be the
natural frequency of resonance as determined by the physical dimensions. If
the applied alternating electric field is off-resonance then very weak
mechanical vibrations will be set.
As seen in Figure 4.6, there is no polarization under zero strain. When
tensile or compressive strain is applied to such a crystal, it alters the
separation between (+)ve and (-)ve charges in each elementary cell. This
leads to a net polarization in each unit cell at the crystal surface. The
Polarization is proportional to the applied Stress and its polarity is direction
dependent. In the Figure 4.6, Elongation or tensile strain causes positive
polarization and Compressive strain causes negative polarization.
Stress creates Electric Field and Electric Field creates Elastic Strain causing
the physical dimension to alter in accordance with the electric field.
Besides Quartz, Rochelle Salts, Tourmaline, Sodium Potassium Tartarate
we have piezoelectric ceramics. PZT is an example of piezo-electric
ceramics. PZT is polycrystalline ferroelectric material with PEROVSKITE
crystal structure.
Perovskite has Tetragonal/Rhombohedral structure very close to Cubic
Structure. They have a general formula as follows:
A®* + B3* + OF — ABO;
Here A is trivalent metal ion such as La.
B is trivalent metal ion such as Al.
Rhombohedral Perovskite is LaAlO3.
PZT is a mass of minute crystallites. Above Curie Temperature it exhibits
simple cubic structure with (+)ve charge center and (-)ve charge center
being coincident as shown in Figure 4.7.a. Hence the crystal is centro-
symmetric with no permanent dipoles. It is found to exhibit paraelectric
behavior. Ability for undergoing electronic polarization as discussed in
Physics of Dielectric Chapter.
Paraelectricity is the ability of many materials (specifically ceramic
crystals) to become polarized under an applied electric field. Unlike
Ferroelectricity; this can happen even if there is no permanent electric
dipole that exists in the material, and removal of the fields results in the
polarization in the material returning to zero. The mechanisms which give
rise to paraelectric behaviour are the distortion of individual ions
(displacement of the electron cloud from the nucleus) and the polarization
of molecules or combinations of ions or defects.
Paraelectricity occurs in crystal phases in which electric dipoles are
unaligned (i.e. unordered domains that are electrically charged) and thus
have the potential to align in an external electric field and strengthen it. In
comparison to the ferroelectric phase, the domains are unordered and the
internal field is weak.
The LiNbO3 crystal is ferroelectric below 1430 K, and above this
temperature it turns to paraelectric phase. Other perovskites similarly
exhibit paraelectricity at high temperatures. Paraelectricity may provide an
alternative to the traditional heat pump. A current applied to a paraelectric
material will cause it to cool down - which could be useful for refrigeration
or for cooling computer chips.
(a)
(b)
.
os +
Go tT!
Figure 4.7. Crystal Structure of PZT above and below Curie Temperature.
As shown in Figure 4.7.b. below Curie Temperature it takes tetragonal
symmetry. Each Unit Cell has built-in electric dipole which may be
reversed by the application of electric field. These electric dipoles may be
re-oriented in any desired direction by the application of appropriately
directed electric field. This is analogous to Ferro-magnetism hence it is
called Ferro-electricity.
In PZT, below Curie Temperature , there are domains of polarization known
as WEISS DOMAINS as shown in Figure 4.8.a. Within WEISS DOMAIN,
the dipoles are self-aligned hence WEISS DOMAINS has a net polarization
measured by dipole moment per unit volume. But WEISS DOMAINS in
PZT are randomly oriented as shown in the Figure 4.8.a.Overall
polarization or Piezoelectric effect is zero. But this mass of minute
crystallites in PZT containing randomly oriented WEISS DOMAINS can be
induced to have net polarization by the application of strong electric field.
This is called Electric Poling and has been illustrated in Figure 4.8.b.
(a) Randomly oriented Weiss Domains
(b)Aligned Weiss Domains by Electric Poling.
Figure 4.8. PZT material below CURIE TEMPERATURE before electric poling and after poling.
As seen in Figure 4.8.b. after Electric Poling all Weiss Domains are forced
to be oriented in a given direction by the application of Electric Field in the
desired direction below Curie Temperature. The domains most nearly
aligned with the applied electric field grow at the expnse of the other
domains. Even after field is removed the alignment remains locked in the
desired direction, giving PZT a remnant electric polarization and a
permanent deformation making the material anisotropic. In anisotropic
materials the material property depends on the direction of measurement.
This is exactly as in Ferro-magnetic Materials. Just as we have B-H
Hysteresis Loop in Ferro-Magnet we have Polarization-E Hysteresis Loop
in Ferro-electric Materials as shown in Figure 4.9. D closely follows
Polarization-E curve.
Remnent Polarization for soft PZT is P, = 0.3 [(C-m)/m?]= 0.3C/m?=Dipole
Moment per unit Volume.
In piezo-electric materials we have Mechanical Deformation versus
Applied Electric Field curve. This also shows a hysteresis loop showing
plasticity and plasticity loss.
Saturation Polarization
B(Weber/m*2) P(dipole moment/cc) y
Pr
(rement polarization)
Remnent Magnetization .
in Ferro-Magnetic Material.
H(Tesla) E (V/cm) ——-
—_—— .
Pr
(remnent poarization
under negative field).
(saturation Polarization) undernegative applied field.
(a)B-H Hysteresis Loop. (b) Polarization-E Hysteresis Loop.
Figure 4.9. Polarization-E curve is analog of B-H Hysteresis Loop.
In Table 4.5.1. the main electric parameters of standard Piezo-electric
materials are given.
Table 4.5.1. Dielectric Constant and Q-Factor of Quartz, PZT 5A and
PZT 4A.
PZT PZT
Materials Quartz 5A(NAVY ID) 4A(NAVY II)
PZT PZT
Materials Quartz 5A(NAVY I) 4A(NAVY II)
Dielectric 4.5 1800 1300
Constant
4
Q-factor He 80 600
Section 4.5.1.Applications of Piezo-electric and Ferro-electric
Materials.
Quartz are used as high Q, high precision Mechanical Resonators and find
wide applications as generating stable electric oscillations for Watches,
Clock Waveforms in Computers and for generating Carrier Waves in Radio
Broadcast Stations.
To date Quartz Crystal Oscillators (Xtal Oscillator) is the stab lest
Frequency Generators. The Frequency of Xtal Oscillator does not drift with
temperature , aging or with varying load. The Resonance Frequencies or
Natural Frequencies are well defined by the physical dimensions of the
crystal and oscillation occurs at natural frequencies. Quartz Xtal Oscillators
are very small. It consists of a thin piece of Cut Quartz Wafer with two
parallel surfaces metalized to make required electrical connections. The
physical dimensions of the crystal are critical in faithfully producing a
given frequency.
PZT are generally used as actuating systems in which they operate they
operate below natural resonance frequencies and in which the ability to
generate high forces and high spatial displacements is more important e.g.
in high performance Ultra-sonic Transducers. PZT can be shaped in any
fashion and it can be polarized in any direction.
Chapter 5_Introduction-Nanomaterials.
Chapter 5 and its sections give the definition of nano-science and nano material, gives examples of nano-materials
and its external manifestation. Subsequently the classification of nano-materials and formation of stable nano-
structure.
Chapter 5_Introduction-Nanomaterials.
The Universe ranges from 10° cm to 10°"cm. The first Length is the distance from our Earth to the observable
Event Horizon of the Universe which is 13.8blightyears(ly) = 4.23313 b parsec(pc) = 1.30622*10*°m. The second
length is Planck’s Length:
hG ao
l= |g = 1.616199(97) x 10°Fm
Planck’s Length sets the fundamental limit on the accuracy of length measuremment.
As proto-human-society and human society has developed, its capabilities have developed. Its tool materials
improved from flint stones to copper to bronze to iron implements to steel to polymers to Silicon and finally to
nano-materials. In Table 5.1, we show the development of Proto-human society due to anatomical changes in
homo-species. Once modern humans emerged human society has developed both due to the development of
relation of production and due to the development of tools of production.
Relation of production has developed from primitive communism to slavery to feudal and to the capitalist society.
The modern Capitalist Society is experiencing the labour pains for delivering a Socialistic Society.
The scale of production has evolved from Small Scale Owner Managed Production to Large Scale Corporate
Managed Production.
Table 5.1. Stages of development in Proto Human Society and Human Society based on anatomical changes,
based on relation of production and based on the development of means of production.
Relation of Means of
Date(ya)! Anatomical? Production Materials used Production
AM AustralopethicusA farensis Savage & Primitive Primitive feo no Roots & fruit
communism hand grip~. gatherer
rare Choppers F
2.5M Homo-habilis(handy man) Say eee Paine &Flakes, Oldwan Fomeie .
communism scavenging
Style
Discovery of fire
nae: Refined,
1.9M Homo-erectus(erect man) Pri Be ; precisionGrip hunting
rimitivecommunism
features developed
Date(ya)!
800,000
200,000
70,000(last
ice age
begins)
50,000
50,000To
40,000
40,000
to30,000
10,000
9,000
Anatomical?
Homo-sapiens(intelligent
man)
Moder humansEmerge in
AFRICA
Migrate out of Africa and
replace all primitive
homo-species.
Modern Humans
reachAustralia and get
isolated as Australian
Aborigines.
Migration to Asia
In;land migration from
Asia to Europe
Modern humans push
toCentral Asia and arrived
in the grassy steppes of
Himalaya
From S.E.Asia and China,
migration to Japan &
Siberia took place.
North Asians migrated to
N.America via land
bridge across Arctic from
Siberia
Great delugeModem
Humans
Modem Humans
Relation of
Production
Barbarians &
Primitivecommuinsm
Barbarians &
Primitivecommuinsm
Barbarians &
Primitivecommuinsm
Barbarians &
Primitivecommuinsm
Barbarians &
Primitivecommuinsm
Barbarians &
Primitivecommuinsm
Barbarians &
Primitivecommuinsm
Barbarians &
Primitivecommuinsm
Barbarians &
Primitivecommuinsm
Barbarians &
Civilized Society-
Private Property,
Monogamous
Family, State
Slavery
Materials used
Precision Grip
matured & tear
shaped hand
axesDeveloped,so-
calledAcheulean
style
STONE AGE,,
cave-dwelling,
refined flint-stone
tools
Stone tools and
weapons
Stone tools and
weapons
Stone tools and
weapons
Stone tools and
weapons
Stone tools and
weapons
Stone tools and
weapons
COPPER AGE
Native Copper
used.Smelted
Copper used,
BRONZE AGE
Means of
Production
hunting
hunting
hunting
hunting
hunting
hunting
hunting
hunting
hunting
Hunting &
agriculture fc
fodder
Mining,
Agriculture fi
grains, artisal
Relation of Means of
Date(ya)! Anatomical? Production Materials used Production
6,000... Mining,
5,500... Modern Humans Feudalism Silver,Tin, Bronze. Agriculture,
5,500... artisans
3,400
tatapoee:: || = enes ears IRON AGE
Industrial
1800CE to er Revolution &
1940CE Modern Humans Capitalism STEEL AGE Large Scale
Production
1940- se Factory
1960CE Modern Humans Capitalism POLYMER AGE producto
Computerizal
1960CE to Viodera: Feainans Automation,
present SILICON AGE Robotization
Miniaturizati
Miniaturizati
1980 to
present Modern Humans NANO AGE & System
Integration
1. Ya = years ago.
2. Hand — grip = hand curves to grip the tool and stiffens at the base of the thumb and in the mid-hands to
stabilize the hand and dissipate the force. This complex of traits gives dexterity in tool making and tool
handling.
3. “When Early Hominines got a Grip”, Meeting Briefs, American Association of Physical Anthropologists, 9-
13 April, 2013, Knoxville Tennese, Science, Vol. 340, 26h April 2013, pp 426-427.
What is Nano-materials ?
Nanoscience = Study of nanoscale materials, properties and phenomena.
Nanoscale Materials = Specifically: < A material 100 nm along one dimension (Out of three dimention.
Nanotechnology = Applications of nanoscience to industry and commerce.
"Wet" nanotechnology, which is the study of biological systems that exist primarily in a water environment.
"Dry" nanotechnology, which derives from surface science and physical chemistry, focuses on fabrication of
structures in carbon (for example, fullerenes and nanotubes), silicon, and other inorganic materials.
Computational nanotechnology, which permits the modeling and simulation of complex nanometer-scale
structures.
5.1. Examples of Nanoscience.
The blue color of the butterfly shown in Figure 5.2 is due to Nature’s remarkable nanotechnology — the blue color
is known as “structural color” (i.e. no pigment) that originates from the nanostructure in the dorsal wing. The
structure interferes with certain wavelengths of light to produce an beautiful . The underside of the wing , on the
other hand is brown. Why?
Figure 5.3. Nanostructure in Butterfly’s wings produces the interference fringes.
Three setae of
morpho wing inter-
acting with light are
shown.
Such layered
structures are
found in many
insects that display
iridescence.
The Urania moth,
for example, has a
multi-layer with a
series of parallel
plates of specified
thickness and
spacing.
Figure 5.4.The layered structure in Butterfly’s Wings, through alternate constructive and destructive
interference, gives rise to coloured fringes.
In early morning , dew drops curl up to form pearl beads on the lotus leaf in a pond. This is an example of
nanoscience. The micro-nanostructure of the lotus leaf exhibits super-hydrophobicity. Superhydrophobicity means
repulsion of Lotus Leaf towards Water Dew Drops. Hence dew drops early in the morning curl up as pearl beads.
The micro-nano
structure of the lotus
leaf exhibits
superhydrophobicity
When contact with a
solid substrate is
minimized in this way,
water tends to behave
as if it were on a non-
polar surface and
beads up.
Figure 5.5. The water beads formation on lotus leaf is a beautiful example of superhyrophobicity.
5.1.1.Self cleaning in Morpho Aega.
Figure 5.6. The attraction to water is higher than the static friction force between the dirty particle( marked
in red) and the pointy surface, the dirty particle will be absorbed by the droplet.
Morpho Aega, a species of Butterfly, has the ability to self clean its wings based on nano-sceience. In Figure 5.6,
water-droplets are shown in blue. These water drops roll-off the wing in a “radial out” direction from the central
axis of the butterfly body due to directional adhesion of the super-hydrophobic wing. The direction of rolling is
tuned by controlling the air-flow by the posture of the wings. The water droplets adsorb the dirt particles by
electro-static attraction and these droplets carry away the dirt as the droplets roll off.
5.2. Classification of Nano-materials.
Materials
Amorphous Crystalline
Superstructure Quasicrystals
Nano-materials are classified in exactly the same manner as Bulk Materials. The crystalline materials have some
standard Unit Cells (Cubic Cells, FCC, BCC, Hexagonal Unit Cell, tetrahedral structure) which are repeated in all
3-Dimensions.
A Single Crystal of nanometer size less than 100nm is referred to as nano-crystal. It is a single domain crystal with
the diameter of the single domain less than 100nm. In Figure 5.7 the nano-crystals are shown.
Quasicrystal
Superstructure
Figure 5.7. A single-crystal of 100nm size or less is a single domain crystal. It is Quasicrystal if in pyramid
form. It is Crystalline if in cubic form. It is super structure if single domain crystals are arranged in a
regular periodic form.
5.3. Formation of Stable Nano-structure.
A stable nano-structure is the minimum energy configuration and a minimum energy configuration is obtained in
the following manner:
1+6=7 balls.
Figure 5.8. One atom surrounded by 6 atoms gives minimum energy configuration hence stable
configuration. So a monolayer is stable when we have 7 atoms.
In bulk-materials, minimum energy configuration is spherical. A sphere is a minimum energy configuration from
hydro-static equilibrium condition. But this is not true for nano-materials.
In nano materials first stable configuration has 13 atoms as shown in Figure 5.9.
Through inspection it can be shown that stable configuration is achieved by having;
M*(K) atoms = (1/3)(10K?+15K2+11K+3) for K" configuration where K = 1,2,3.....
For K= 1, first stable nano-crystal has 13 atoms.
For K= 2, second stable nano-crystal has 55atoms.
For K= 3, third stable nano-crystal has 147 atoms.
For K= 4, fourth stable nano-crystal has 309 atoms.
For K= 5, fifth stable nano-crystal has 561 atoms.
These configurations are shown in Figure 5.10.
Figure 5.9.A 3-D nanocrystal is stable ewhen there are 6 atoms surrounding the core atom and 3-atoms from
the top and 3-atoms from the bottom. Meaning by when we have 13 balls then we have first stable 3-D
nanocrystal structure though it is not a spherical structure.
Figure 5.10. Stable nano-configuration corresponding to K = 1,2,3,4 and 5 are shown.
In Figure 5.11. it is shown that the surface atoms dominate the crystal structure as we move from Bulk
configuration to nano-particle configuration. Because of this dominance by surface atoms in nano-crystal
configuration, material properties of nano-particle is completely different from those of the bulk crystal.
Surface to volume ratio:-A 3 nm iron particle has 50% atoms on the surface
-A 10 nm particle 20% on the surface
-A 30 nm particle only 5% on the surface
Full-shell Clusters Total Number Surface Atoms
of Atoms (%)
1 Shell eS 13 92
2 Shells eS 55 76
3 Shells ee 147 63
4 Shells 309 52
5 Shells 561 45
7 Shells 1415 35
Figure 5.11. Nano-particle structure for K th Stable, Minimum Energy, Configuration for K = 1, 2, 3, 4,5
and 7.
100
asaene
aeoneene
=
Bulk Atoms
With
Particle Size(nm)
% of Atoms in Bulk/on Surface
Figure 5.12. Calculated Surface Atoms to Bulk Atoms Ratio for Solid Metal Paricles vs the Size of Particle
(nm). [Curtsey: Kenneth J.Klabunde, Jane Stark, Olga Koper, et.al. “Nanocrystal as Stoichiometric Regents
with Unique Surface Chemistry”, Journal of Physical Chemistry, Vol.100, pp. 12142-12153,(1996)]
As the particle size changes from Bulk — Size to Nano - Size, the overall structure changes and Valence electrons
become de-localized. What does this de-localization mean?
We have seen in Band-Theory of Solids that when atoms are far apart the orbital electrons donot interact and the
Energy Difference between two consecutive states is at the maximum. But when they are brought close together
they start interacting and band gap between consecutive band reduces. What this means that nano-size particle will
have a larger band-gap and as nano-size increases to bulk-size, band-gap asymptotically approaches the bulk band-
gap. This leads to different physical and chemical properties continuously graded as size increases from nano to
bulk size.
The following properties are affected by nano to bulk-size transition:
Optical properties, Bandgap ,Melting point ,Specific heat, Surface reactivity, Magnetic property
and Electrical conduction.
Section 5.4. Nano-Size Effect on various opto-electronic-magnetic properties of nano
materials.
Section 5.4 describes the nano-size effect on the band-gap, specific heat, melting point,
colour of the light emitted from Quantum Dot, on the charging of a nano particle and the
effect of confinement on density of states.
Section 5.4. Nano-Size Effect on various opto-electronic-magnetic properties of
nano materials.
A nano-particle or a Quantum Dot Semi-conductor ‘Si’ is a cluster of large number of
atoms arranged in the minimum energy configuration. If the there are N atoms there are
about N electrons or a few times more in this cluster. Most of these electrons are tightly
bound to their host atoms. But there are few carriers electrons and holes which behave
like particles in an infinite potential well. The scenario is depicted in Figure 5.12a.
These behave just as electrons behave in isolated Hydrogen Atoms. Hence Quantum
Dots are Artificial Atoms.
n = principal
quantum number
Figure 5.12a. A particle in an infinite Quantum Well of One Dimension
The Schrodinger equation of a particle in an infinite potential well is:
Hw = Ew where H is the Hamiltonian Operator
In Quantum Mechanics we have canonical variables: x and p or t and E.
Linear Momentum ‘p’ is equivalent to an operator:
iia nd E n?
3 —ih— — th—
P ax not
Therefore Schrodinger Equation for a particle in an infinite potential well is:
2 2 2
ww -(2 +v) —— xoS ty = BY 5.41
2m,
Hamiltonian Operator operates on Matter Wave w to yield the eigen value of energy E.
(5.4.1) simplifies to Second Order Ordinary Linear Differential Equation with two
arbitrary constants determined by the two boundary conditions namely:
y(L) = (0) = 0
The simplified Schrodinger Equation is:
op , 2me(E-V), _
Pm) + -— w=0 5.4.2
The Solution is Complementary Function:
2m,(E —V
w(x) = BExp[—iax]+ CExp[+iax] wherea= ame) 5.4.3
With the given two boundary condition (5.4.3) simplifies to:
w(x) = DSin[ax] 5.4.4
To satisfy the given boundary condition:
_ 2m,(E—V) _ _ nah?
aL=nn orL Or (E-V)= 2m, 5.4.5
If V = 0 then the discrete energy states occupied by a carrier in an infinite 1-D Potential
Well are:
n?* 77h?
E,, = =n'E, 5.4.6
2m,L?
The energy separation between two adjacent energy states is:
= 5.47
1 2m,L? io
The actual energy states in a semiconductor are:
2%2
mh
Eo? = pBulk 4 —___ 5.4.8
g g 1 2m,L
In Equation 5.4.8., L is the Radius of the Quantum Dot. By adjusting the size (i.e. R) of
the Quantum Dot, it is continuously tunable through all the seven colours of a Rainbow.
The delocalization of electrons and structural changes in nano materials causes band-gap
energy change in semi-conductor nano materials as well as changes in the following
properties:
i.Optical properties,
ii.Melting Point,
iii. Specific heat,
iv.Surface reactivity,
v. Magnetic Properties,
vi.Electrical conduction.
Table 5.2. gives the Specific Heat of Palladium, Copper and Rubidium in bulk-size and
in nano-size.
Table 5.2. Specific Heat of Pd,Cu and Ru in bulk and in nano-size.
Elements Bulk(J/(mol.K)) Nano(J/(mol.K)) ae - %increase
Pd 25 ay 6nm 48
Cu 24 26 8nm 8.3
Ru 23 28 6nm 22
As the bulk-size moves to nano-size, the Melting Point reduces as given below:
2T,o
A@ = where A@
r
= deviation in M.P. from the bulk value and it can be as large as 100°C
for particle size r~10nm and T, = Bulk M.P.and o = surface tension coef ficient for
liquid — solid interface and p = particle density and r = particle radius and L
= latent heat of fusion.
In Figure 5.13 it is shown that M.P. of gold particles decreases dramatically as the
particle size gets below 5nm.
)
)
90
80
700
600
Melting Point [°C]
3
0 1 2 3 4 5 6 7 8 9 10 11
Particle Radius [nm]
Figure 5.13.M.P. of Gold Particles vs Particle Size.
Blue light is used to excite Quantum Dots of reducing size. Progressively shorter
wavelengths are emitted as the dot size decreases. The reason is explained in the
Vue
GOe?:
Figure 5.14. Quantum Dots of smaller size emit shorter wavelengths.
QD comprises of million atoms and equivalent valence electrons. But QD does have a
few free electrons. These electrons behave like particle in an Infinite Potential Well. The
Schrodinger Equation has been solved in m33488. That solution tells us electron
occupies discrete energy states given by the following equation:
2p2
n“h
E, = a where W = 2r(radius of the dot) 1
h2
Equation 1,explicitly tells that Energy Gap = ——— 2
8SmWw?2
From (2) it is clear that smaller the dot size larger, larger will be the energy packet
absorbed during excitation or larger will be the energy packet radiated when Dot settles
to ground state. This is the reason why in Figure 5.14 as the dot size decreases it is seen
to emit shorter wavelengths This emission can be LASER type coherent emission or
LED type incoherent emission.
Confining a carrier in atleast one spatial dimension at the scale of the order of de Broglie
Wavelength leads to Quantum Size Effect.
In Bulk - 3 degrees of freedoms with 0 degreeof Quantum Confinement. So we get
bands of permissible energy.
In Quantum Well — 2 degrees of freedom with 1 degree of Quantum Confinement. .
In Quantum Wire - 1 degree of feedom and 2 degrees Quantum Confinement.
In Quantum Dot — 0 degree of freedom with 3 degrees of Quantum Confinement.
Hence in a Quantum Dot, electron behaves as an electron in infinite 1-D Quantum Well.
Hence electron behaves as it behaves in a Hydrogen Atom and carriers have discrete
energy states. Therefore QD is referred to as an artificial atom.
QD are extremely small semiconductor structures ranging from 2nm to 10nm. At these
small dimensions, materials behave differently giving QD unprecedented tunability and
enabling never before seen applications in Science & Technology.
The Band-gap of nano-structure:
h?n7/ 1 1
Enano — Ebulk (— —) 3
* ot 2R? a
QD LEDs can produce any colour including white light, it is extremely energy efficient
giving the highest Lumens per Watt (uses only few watts) whereas CFL uses more than
10W. Its lifetime is 25 times that of incandescent lamp. But these are expensive . As its
price falls due high demand and mass-production, these will become affordable.
Section 5.4.1. Electrical Conduction.
In bulk metal, current flow is according to Ohm’s Law:
V pL 1
= R where R=— andp=-= where q = electronic charge,n
o QH,n
= density of conducting electron and u,, = electron mobility
qa
= ———— where v,;
AmegM Vp
= Fermi Velocity and is calculated from the following relationship:
™
3
5™03 = (Er Fe)
5
In nano-material, band theory does not hold good so Ohm’s Laws is no longer applicable
and special transport equations have to be formulated for accounting for the tunnel
currents from nano-particle to nano-particle.
Section 5.5. Electric and Magnetic Properties of matter.
Electron is responsible for two distinct properties:
1- Charge -> It gives all electrical properties of mater.
Conductivity, Paraelectric, ferroelectric, antiferroelectric, diaelectric etc.
2- Spin- > It gives magnetic properties.
Paramagnetic, Ferromagnetic, Antiferromagnetic, Diamagnetic, superparamagnetic,
Ferrimagnetic, Canting magnetism, Superferromagnetism.
5.9.1.Charging Energy of nano-particle.
Capacitance of a Sphere is:
C = 41) ¢,(a) where a = radius of the sphere. +
Therefore charging energy U for nanoparticle of radius a is:
2 2
q_ 4
This is 10 times larger than thermal energy at Room Temperature of 300K which is
25meV.
25meV corresponds to the charging energy of a nano-particle of radius 14.4nm.
So we conclude there is considerable charging energy for nano particle.
5.5.2. Density of States.
It can be shown that Density of permissible Energy States is as follows:
1 2m
= —_ (—_)3/2 V
D(E)=V a st =) XVE for 3D Bulk 5
1 (2m
D(E)=Vx— (=) for 2D Quantum Well 6
2n \h?
D(E)=Vx— sr)? x (=) for 1D tum Wi 7
= = G2 ) TE or 1D Quantum Wire
Pictorially it is shown in Figure 5.16
3D
2D
De)
1D
—
Figure 5.16. Density of States vs Energy for 3D Bulk, 2D Quantum Well and
1D Quantum Wire.
EVEN SEMESTER2014-MidSemester Examination Answers
This gives the model answers of Mid Semester Examination conducted for 2014 EVEN SEMESTER. Electrical
Electronic Materials.
Even Semester Mid-Semester Test
Electronic & Electrical Materials EC1419A
Total Marks:30 Duration: 2hrs
Question 1 is compulsory and answer any two out of the remaining three.
Question 1.[15 points]
a
re
id
ad
e
ee
ad
r
a
ad
rt
I.What are the MKS units of mass, velocity (v), acceleration (a), force, momentum (p), energy ? Answer: Kg,
m/s, m/s2, Newton, Kg-m-s-1, Joules.[1/12+1/12+1/12+1/12+1/12+1/12=1/2Point]
ILE = Eoexpl[j(@t — kz)]_ This is an equation of a forward travelling wave. Define a, t, k and z. What is the
velocity of propagation ? Answer: w=radians/s, t=second,k=1/m, z = direction of propagation. Velocity of
propagation= o/k (m/s).[1/10+1/10+1/10+1/10+1/10=1/2Point]
If. w = Woexplj(@t-kz)]_This is also a travelling wave. What is Wg and what is | w*y | ? Answer: W0 is
Probability Amplitude, | y*y | is Probability Density.[1/4+1/4=1/2Point]
IV. J = I@ = n[h/(2n)]. This is Bohr’s Law of electrons orbiting the nucleus. Define all the terms.
Answer:J — angular momentum, I - moment of inertia, @ — orbital angular velocity, n> principal
quantum number, h > Planck’s Constant.[1/10+1/10+1/10+1/10+1/10=1/2Point]
. V. Define the four Quantum Numbers n, L, m,s ? Answer: n — Principal Quantum Number,
L- Azimuthial Quantum Number, m > magnetic Quantum Number, s > Spin Quantum Number.
[1/8+1/8+1/8+1/8=1/2Point]
VI. What is COORDINATION NUMBER=N. Si, Ge have DIAMOND Crystal Structure and GaAs is ZINC-
BLENDE Crystal Structure. In both crystal structures N = 8. Justify this statement ? Answer: N > number of
atoms belonging to each Unit Cell of the crystal structure. In Diamond structure there are (8corner
atoms)x(1/8)+(6 face centered atoms)*(1/2)+(4 body centered atoms)= 8 atoms per unit cell.
[1/8+3/8=1/2Point]
VII. Number density = N* = 8/a?. What is ‘a’ ? Answer : ‘a’ is the Lattice Parameter.[1/2Point]
VUI. For Si,Ge and GaAs ‘a’ = 5.43A°, 5.646A° and 5.6533A° respectively. What is A° ? Answer: A° is
Angstrom =10-10m = 10-8cm .[1/2 Point]
VIV. Weight of one ------------ _= (AW gm/mole)= (Nayo atoms/mole). Fill up the BLANK . Answer: ATOM
[1/2 Point]
X. Wt of one atomx N* =p. Define the terms ? Answer: N* > number density, p > weight density.
[1/4+1/4=1/2Point]
XI. n HYDROGEN ATOM, electron occupies discrete energy states given by E= -13.6/n*. What is the unit
of energy and what is ‘n’ ? Answer: Unit of energy is eV and n- Principal Quantum Number.
[1/4+1/4=1/2Point]
XII. Hydrogen Atoms produce Lyman Series[2 > 1,3 > 1,4 1,5 1,6- 1], Balmer
Series[3 > 2,4 > 2,5 > 2,6 > 2,7 > 2] and Paschen Series [4 > 3,5 > 3,6 > 3,7 > 3,8 — 3] of Spectral lines. What
do the square bracket terms mean? Answer : Square Bracket terms imply transition from Excited States
to Ground States.[1/2Point]
i. XIU. FCC has 74% parameter and BCC 68% parameter. What is this parameter and what is the full form of
FCC and BCC? Answer: FCC is Face-Centered-Cube and its packing density is 74% and BCC is Body-
Centered-Cube and its packing density is 68%.[1/6+1/6+1/6=1/2Point]
a Type Orientation
45° N <111>
90° P <100>
180° N <100>
0° P <111>
i. XIV. What does the above Table refer to ? Describe the meaning of the terms entered in the boxes ? Answer:
The angle between the Primary Flat of the Si Wafer and the Secondary Flat of the Si Wafer is ‘a’. Type
tells if Si- Wafer is N-Type or P-Type. Orientation gives the cleavage plane Orientation in Miller
Indices.[1/6+1/6+1/6=1/2Point]
zo 82\ 2m*
XV. What do the different terms refer to in the above Integral Equation? Answer: N(E)=density of states per unit
volume per unit energy, P(E) — Fermi-Dirac Statistics or the probability of occupancy at energy E by
Fermions. dE — elemental energy, m* — effective mass of electron in the conduction band.X0 — Fermi Level
in Metal with respect to the bottom edge of the Conduction Band i.e.X0=EF- EC.
[1/10+1/10+1/10+1/10+1/10=1/2 Points]
XVI.
n= GD
In the adjacent equation and in Question XV, Xp is the same parameter. What is this parameter.Answer: XO=EF-
EC.[1/2 Point]
XVII.
E,(?) _ 12
4@)’
What are the different parameters and what are their units ?Answer: Eg is Band-Gap in eV and A is the
wavelength of the photon emitted accompanying the radiative transition from Conduction Band to the
Valence Band.[1/4+1/4=1/2]
XVIII. Define the parameters and give the units ?
Semiconductor E, nj €,(DC) ? p;(300K) Nc Ny
Semiconductor Ey, nj €,(DC) ? p;(300K) Nc Ny
Si 1.12 1x1010 11.7 5x1022/cc 3.2x10° 3.2x10/ec 1.8x
Ge 0.67 2x1018 16.2 4.4x102/cc 48 1x10!%ec 5x1
GaAs 1.424 2.1108 12.9 4.42x1022/cc 3.3x108 4.7x10!7/cc 7x1
Answers: E,(eV) , ni(#/cc), €(DC)rel.permittivity(Dimensionless), #density of atoms in the crystal ,
p\(300K)Intrinsic Resistivity in Q-cm, Nc_Effective Density of States at EC,Ny_ Effective Density of States
at EV [1/14+1/14+1/14+1/14+1/14+1/14+1/14=1/7]
XIX.
Varige = ME where =
QTscattering
My
Define the terms and give the units. v = 10’cm/s. What is this velocity. Answers: vdrift = electric drift
velocity(m/s), ja(mobility) (cm2/(V-s)), q=electric charge(Coulombs), tscattering(Mean Free Path or
Relaxation Time)(sec),m*e=effective mass of electron(Kg)[1/28,1/28,1/28,1/28,1/28,1/28,1/28,
1/28, 1/28,1/28, 1/28,1/28,1/28,1/28,=1/2]
XX.
1 1
=— + :
Heffectve Himp Hiattice
This rule is called Matthiessen’s Rule. Define the parameter and give the units of the
parameter.Answers:peffective, — effective mobility, pimp, — mobility due to impurity scattering,
plattice, — mobility due to lattice scattering, cm2/(V-s), [1/8+1/8+1/8+1/8=1/2]
XXL.
3 7/2
Hiattics © z a/2 and Limp «
(Na+Na)
What are these power laws.Answers: lattice scattering inverse 3/2 power law , impurity scattering 3/2 power
law [1/4+1/4=1/2 points]
XXII. o = qun Define the terms and give the units.Answers: o — conductivity(Siemens/cm), q — electronic
charge(Coulomb),p — mobility(cm2/(V-s),n > conducting electron number
density[1/16+1/16+1/16+1/16+1/16+1/16+1/16+1/16=1/2 Points]
XXIII. What special information is given by this Table.
Metal Au Cu Ni Pt
Effective Mass/Free Mass of electron 1.1 1.01 28 13
Answers: Ni and Pt are heavy Fermionic Metals [1/4+1/4=1/2]
XXIV. What is the basic difference about electron transport reflected by the Table below ?
Element Ep t(femtosec) L(Angstrom) Wr Vel(x10° m/s)
Li 4.7 9 90 9.96
Na 3.1 31 250 2.3 8.08
K 2.1 44 293 2.2 6.55
Cu 7.0 27 328 12.15
Ag 5.5 41 441.6 10.77
Ge FysEe/2 2217 2106 1.16
Si EytEg?2 767.6 729 1.16
GaAs EytEg?2 4890 4645.5 1.16
1. This velocity can be thermal velocity or Pauli Velocity.
Answers: Conducting electron is strongly scattering in metallic crystalline lattice but very wekly scattering
in Semiconductor as exhibited by short Mean Free Path in metals and by long Mean Free Paths in Semi-
conductors.[1/4+1/4=1/2]
XXV. In the Table given in Question XIV which is Pauli Velocity and which is Thermal Velocity?
Answers: Conducting Electron velocities in Metals are Pauli velocities because they arise out of Pauli-
Exclusion Principle whereas conducting electron velocities in Semi-conductor are thermal velocities.
[1/4+1/4=1/2]
XXVI. In Question XVIII, intrinsic Ge has the least resistivity and intrinsic GaAs has the maximum , Why?
Answers; Ge has the least Band-Gap and GaAs has the widest Band-Gap.[1/4+1/4=1/2]
XXVII.
1
P(E) = 1 Exp }
This is what statistics and does it tell. At T= 0 Kelvin what kind of distribution is it?Answers: This is Fermi-
Dirac Statistics, P(E) gives the probability of occupancy of E by Fermions, and Absolute Zero the
distribution is RECTANGULAR.[1/6+1/6+1/6=1/2 Points]
XXVIII.
M(E) = 4) 97(E — Ec)¥?
What is this expression ? Define all the terms.Answers: N(E) — Density of States per unit energy, (E-EC)
~ kinetic energy of conducting electron in conduction band.[1/4+1/4=1/2]
XXIX. If Ultra-Violet light of A = 0.2p:m is incident on a metal then which of the metals listed in the Table below
will respond and emit electrons ? Answers;UV photon has 6.2eV hence it is powerful enough to cause photo-
emission in all the Metals listed in the Table below.All the metals have a work-function less than 6.2eV.[1/2
Poiint]
Metal Work-function Metal Work-function
Na 2.3eV Ca 3.2eV
K 2.2eV Ba 2.5eV
Cs 1.8eV Pt 5.3eV
W 4.5eV Ta 4.2eV
XXX
A
n, = NpoExp [-= .What is this expression? Define the terms.
1
Answers: This expression tells how EXCESS CARRIERS decay with respect to time in Semiconductors with
excess carrier life-time or excess carrier relaxation time defined as tm. Excess carriers exponentially decay
with time. If relaxation time is long then excess carriers transient takes a long time to decay.[1/2 points]
Question 2.Describe the step for preparing Electronic Grade Poly-crystal Silicon from sand.
[7.5 marks]
Answer:
Reduction of sand with carbon gives impure polycrystalline Silicon
1
Reaction of pulverized raw silicon with HCI gaseous vapour to form TriChloroSilane
1
Multiple distillation of TriChloroSilane to obtain purified electronic grade TriChloroSilane
L
Thermal decomposition of SiHCI3 at 1000 degree centigrade in Sieman’s reactor to obtain fattened rods of
electronic grade Silicon
Question 3. Balmer Series Spectral Lines from Stars are determined to be at 6563A°,4862A°,4341A°,4102A° and
3970A°. Determine these spectral lines theoretically.
[By the Red-Shift of these Balmer Spectral Lines the velocity of the receding Galaxy is detyermined. The
recession velocity gives the distance of the Galaxy using Hubble’s relationship][7.5 marks]
Answer: A(pim)=1.24/E g (eV) here E g = (13.6eV/4-13.6/n 2 )eV. Since the answer is to be obtained in
Angstrom therefore the final expression is:
Therefore A(Angstrom)=( (1.24/(13.6/4-13.6/n4 2 ))x104 4 )Angstrom.
For n=3 to n=2, A=6564.71A°;
For n=5 to n=2, A=4862.75A°;
For n=6 to n=2, A=4341.74A°;
For n=7 to n=2, A=4102.94A°;
For n=8 to n=2, A=3971.24A°;
Question 4.Detrmine the temperature at which E=Ert+kT will be occupied by electron with P(E)=1%.[7.5 marks]
Answer
This question is wrongly stated. At E=E,+kT the Probability of occupancy is 1/(1+e) =0.2689. It can be never 1%
irrespective of temperature.
The correct question is: Determine P(E + kT) in Fermi-Dirac Statistics. Determine Temperature T at which P(Ep
+ 0.5eV) = 1%.[Answers: 0.27, 1262Kelvin]
You can give either of the answers and fetch full marks.
Chapter 5. Section 5.6. Metamaterials.
Chapter 5, Section 5.6 gives introduces Meta-materials and gives the latest
developments in the field of meta materials.
Chapter 5. Section 5.6. Metamaterials.
Meta material science is a multidiscipline enterprise which includes applied
physics, engineering, material science and nano technology. It aims at
designing materials with physical properties beyond those available in
nature. In metamaterials we want go beyond the natural materials and
fabricate metamaterials with undreamt of properties e.g. invisible cloak.
Metamaterial Science challenges and overcomes the currently held
limitations offered by the classical physics. Careful engineering and mixing
of meta-atoms has led to completely unconventional standards and
yardsticks of classifying the materials.
Metamaterials derive their properties from the structure rather than their
constituent components hence they are called super lattices.
For example Photonic Crystals derive their anomalous properties from
higher order spatial modes whereas meta materials work with dominant
propagation mode and with sub-wavelength spacing between neighboring
elements. Macroscopic desired and tailor made properties can be obtained
by applying several mixing rates and homogenization principles.
Section 5.6.1. Diffraction Limit of Abbe and its conquest by Xu et.al.
Metamaterial Superlens 1
[1. Xu, T.; Agarwal.A.; Abashin, M; Chau, K.J.; Lezec, H.J. “All-angle
negative refraction and active flat lensing of Ultra-Violet Light”, Nature,
497, #7450, pp. 470-474 (2013)]
Ernest Abbe(1840-1905) a top Physicist and the CEO of Carl Zeiss Lns
Company, had set the resolution of a microscope as follows:
kA
2Ay(minimum feature size) = =e where A= wavelength of the image light,n
ina
= refractive index of the space in which the image is formed,k = resolution factor,a
= acceptance angle of the lens system 5.6.1.
According to this Formula, the wavelength of the light and Numerical
Aperture = nSina set the resolution limit.
Finer feature sizes can be resolved if wavelength is shorter and n(refractive
index) is higher.
Present generation of ICs with 40nm node are using Deep Ultra Violet
Light with wavelength 193nm and pure water immersion system to achieve
the node size of 40nm.
Future Generation IC Lithography is going to use Extreme UV of A=
100nm with immersion technique.
In 2000 J. B. Pendry made a proposal of synthesizing Negative Refraction
Index Meta material which could be used as super lens for perfect lensing to
any feature size.[“Negative Refractive Makes a Perfect Lens”, Physical
Review Letter , 85, #18, 39" October 2000].
They suggested that electron cloud in a dielectric behaves like a plasma and
it has the following dielectric theoretical formulation:
e=1-— _here w,, = electronic plasma resonance 5.6.2.
@
When @ < @gp the we have € is negative.
Similarly we can have loops of conducting wire mimicking magnetic
plasma. It wll have the following formulation:
2
@
w=1- =; here w,,, = magnetic plasma resonance 5.6.3.
When @ < @mp the we have J is negative.
Xu et al using Ag and TiO, layers on glass substrate synthesized negative
refraction lens.
Here both ¢€ = -1 and p= -1.
This results in n= +v(pe) . When ¢€ = -1 and p= -1 then n = -V(1€)
Here the characteristic impedance Z = v(\1/e) = Zp = free space
characteristic impedance.
Therefore at the interface of the lens there is no reflection and transmission
is 100%.
Also it is taken flat with a thickness ‘d’.
The negative refraction index restores the phase of the propagating waves
and also the amplitude of the evanescent waves. The device focuses light
tuned to the surface plasma frequency of silver and is limited only by the
resistive losses. Here both propagation wave and evanescent waves
contribute to the perfect resolution of any feature size dimension. Therefore
there is no physical obstacle to perfect reconstruction of the image beyond
practical limitations of aperture and the lens surface.
The field of meta materials which is emerging shows us how to engineer the
refraction of light through metallic composites with nano scale structure so
that we get perfect lensing with no diffraction limit as given by (5.6.1).
Section 5.6.2. The diverse field of Metamaterials.
In their short history, metamaterials have been applied to the most diverse
areas, including invisibility cloaks, artificial optical black holes, cosmology,
high-temperature
superconductors, just to name a few particularly fascinating examples.
It has been used to fabricate Super lens described above.
It has been used in Automotive Industry.
It has been used in THz Time Domain applications.
It has been used in Specrometers.
It has been used invisibilty Cloaks.
In 2002, Lucent Technologies has developed Resonant Antennas.
In 2003, MIT urilized photonic crystals for making metamaterials.
Boeing Company has developed the method for fabricating electromagnetic
materials.
In 2004, Lucent Technologies has fabricated Miniature antennae based on
negative pewrmittivities.
Boeing Company has developed Metamaterial scanning lens antenna
system and methods.
Chapter 7. Magnetic Materials.
Chapter 7, Section 7.1 to Section 7.6 describe Ferro, Ferri, Anti-ferro,Para
and Dia-magnetic materials.
Chapter 7. Magnetic Materials.
[All the FIGURES are at the end of the Chapter]
HISTORY: The most popular legend accounting for the discovery of
magnets is that of an elderly Cretan shepherd named Magnes. Legend has it
that Magnes was herding his sheep in an area of Northern Greece called
Magnesia, about 4,000 years ago. Suddenly both, the nails in his shoes and
the metal tip of his staff became firmly stuck to the large, black rock on
which he was standing. To find the source of attraction he dug up the Earth
to find lodestones (load = lead or attract). Lodestones contain magnetite, a
natural magnetic material Fe304. This type of rock was subsequently
named magnetite, after either Magnesia or Magnes himself.
Earliest discovery: The earliest discovery of the properties of lodestone was
either by the Greeks or Chinese. Stories of magnetism date back to the first
century B.C in the writings of Lucretius and Pliny the Elder (23-79 AD
Roman). Pliny wrote of a hill near the river Indus that was made entirely of
a stone that attracted iron. He mentioned the magical powers of magnetite
in his writings. For many years following its discovery, magnetite was
surrounded in superstition and was considered to possess magical powers,
such as the ability to heal the sick, frighten away evil spirits and attract and
dissolve ships made of iron!
The first paper on Magnetism: Peregrinus & Gilbert Peter Peregrinus is
credited with the first attempt to separate fact from superstition in 1269.
Peregrinus wrote a letter describing everything that was known, at that time,
about magnetite. It is said that he did this while standing guard outside the
walls of Lucera which was under siege. While people were starving to death
inside the walls, Peter Peregrinus was outside writing one of the first
‘scientific’ reports and one that was to have a vast impact on the world.
Earth is a huge Magnet: However, significant progress was made only with
the experiments of William Gilbert in 1600 in the understanding of
magnetism. It was Gilbert who first realized that the Earth was a giant
magnet and that magnets could be made by beating wrought iron. He also
discovered that heating resulted in the loss of induced magnetism.
Interrelationship between Electricity and Magnetism: In 1820 Hans
Christian Oersted (1777-1851 Danish) demonstrated that magnetism was
related to electricity by bringing a wire carrying an electric current close to
a Magnetic compass which caused a deflection of the compass needle. It is
now known that whenever current flows there will be an associated
magnetic field in the surrounding space, or more generally that the
movement of any charged particle will produce a magnetic field.
Birth of Electromagnetic Field: Eventually it was James Clerk Maxwell
(1831-1879 Scottish) who established beyond doubt the inter-relationships
between electricity and magnetism and promulgated a series of deceptively
simple equations that are the basis of electromagnetic theory today. What is
more remarkable is that Maxwell developed his ideas in 1862 more than
thirty years before J. Thomson discovered the electron in 1897, the particle
that is so fundamental to the current understanding of both electricity and
magnetism.
The term magnetism was thus coined to explain the phenomenon whereby
lodestones attracted iron. Today, after hundreds of years of research we not
only know the attractive and repulsive nature of magnets, but also
understand MIR scans in the field of medicine, computers chips, television
and telephones in electronics and even that certain birds, butterflies and
other insects have a magnetic sense of direction.
7.1. Magnetic Circuit — an analog of Electric Circuit.
Just as in an electric circuit , voltage (V) drives current (A) through the
electrical circuit resistance R where R is:
L
= 7 where pis resistivity 2 — cm,L is length of the circuit in cm and A is the cross
V (Volts
— sectional area incm* and I(Amp) = a 7A
R(ohms)
In an analogous fashion, transformer’s core is a magnetic circuit where
Magneto-Motive Force in Amp-Turns(AT) drives magnetic flux in Weber
overcoming the reluctance of the magnetic core where Reluctance is
defined as follows:
L H
R(reluctance) = — where p= Permeabilty (=)
LA ™.
=U, ,A ts the cross
— secvtional area of the magnetic path which in this case is the cross
— sectional area of the transformer core and L is the Length of the magnetic path and
MMF(AT)
R
,, (magnetic flux in Weber) = re
7.2. Ampere’s circuital Law:
Andre Maria Ampere (1775-1836) gave the Ampere Circuital Law.
In the simplest form it is stated as:
H.2nr = I (current enclosed by the circumference of the circle) 7
James Clerk Maxwell (1831-1879) stated it as follows:
6D
V X B = pwJ(electronic current) + 7 (Displacement current) 7A.
(7.4) is Differential Form. In Integral form it is as follows:
¢ B.dl = yl (total electronic Current enclosed within the line integral path) +
= xX A(total area enclosed within the line integral path) At
7.3. Faraday’s Law of electro-magnetic induction.
Whenever the magnetic flux linked with a closed conducting coil changes,
an e.m.f. is induced which causes a eddy current to flow through the coil.
The direction of eddy current is such as to create a magnetic polarity which
opposes the cause of electro-magnetic induction.
If a North Pole approaches the closed coil then eddy current will make the
near face of the coli North Pole so that the approaching Pole is opposed and
if North Pole is receding then near face will act as South Pole so as to
attract the receding Pole.
Mathematically it is:
e.m. f.(indced electro — motive force) = —N——
= —(rate of change of total magnetic flux linkage) 7.6.
Maxwell stated it as follows:
VxE=-— 7.7
p eal == 78
E
7.4. Lorentz Force and Left Hand Rule.
A current in the magnetic field , held transverse to the magnetic field, will
experience a mechanical force in the third perpendicular direction. This
mechanical force is known as Lorentz Force.
Complete statement of Lorentz Force is as follows:
F(Newtons) = q(E+v X B) 7.9.
The first part on R.H.S. is the electro-static force and second is the
magneto-static force on a moving electron. J.J. Thomson used this equation
to determine q/m of an electron in Cathode Ray Experiment. He balanced
the electric force and magnetic force and obtained zero deflection of the
electron beam on the CRO screen.
A line of current I Amperes of length L(m) is held in Magnetic Field of B
Tesla where I and B are transverse to each other then there is a force f
(newtons) acting on the line of current along the third perpendicular
direction. If Current is in the direction of M[i]ddle finger and Magnetic
Field is in the direction of index [f]inger then the Lorentz force is in the
direction of the Thu[m]b as shown in Figure 7.2.
This force F(Newtons)= B.I.L It is this force which causes an armature of a
motor to rotate.
Two parallel lines carrying I amperes each will experience a magneto-static
force of attraction if the current flows are in the same direction and
experience a magneto-static repulsion force if they are in opposite direction.
7.9. Right Hand Palm Rule.
This rule lets us determine the polarity of a solenoid or a coil of current. If
the fingers of right hand clasping the solenoid is in the direction of the
current flow as shown in Figure 7.3. then magnetic flux line will be in the
direction of the thumb.
North Pole of the Solenoid is the end from where magnetic flux lines come
out. The end where they enter the solenoid is defined as South Pole.
7.6. Magnetic Materials.
The materials which are spontaneously magnetized or have a susceptibility
to magnetization are referred to as magnetic materials. They are classified
as:
i.Ferromagnetic- spontaneously magnetized and have strong, positive
susceptibility to magnetization. Ferromagnetic material has been
demonstrated in Figure 7.4. Up to a temperature known as Curie
Temperature, in this case 1000K, spontaneous magnetization is maintained
as shown in the upper part of the Figure but this self alignment of the
dipoles gets disrupted due to thermal fluctuations above 1000K.
ii. Paramagnetic materials: these have no spontaneous magnetization and
have weak, positive susceptibility to magnetization. As shown in Figure
7.9., there is no spontaneous magnetization but application of Magnetic
Field does cause a weak magnetization.
iii. Diamagnetic materials: these have no spontaneous magnetization and
they move away from a region of magnetic field and have a negative, weak
susceptibility to magnetization. Diamagnetism is the property of an object
which causes it to create a magnetic field in opposition to an externally
applied magnetic field, thus causing a repulsive effect. Specifically, an
external magnetic field causes eddy currents in such a way that according to
Lenz's law, this opposes the external field. Diamagnets are materials with a
magnetic relative permeability less than 1.
Consequently, diamagnetism is a form of magnetism that is only exhibited
by a substance in the presence of an externally applied magnetic field. It is
generally quite a weak effect in most materials, although superconductors
exhibit a strong effect.
Diamagnetic materials cause lines of magnetic flux to curve away from the
material, and superconductors can exclude them completely (except for a
very thin layer at the surface).
In Figure 7.6. a comparative study of diamagnetic and paramagnetic
material has been made. In paramagnetic there are atomic magnetic dipoles
but randomly arranged and application of magnetic field can align them to
result in a weak magnetization. But in diamagnetic materials there are no
atomic magnetic dipoles. These get induced by the application of external
magnetic field but opposed to the external magnetic field in accordance
with Lenz’s Law.
In Figure 7.7. we show the magnetization curve with respect externally
magnetic field.
In Figure 7.8, we show the temperature sensitivity of Ferromagnetic and
paramagnetic materials.
B= UH + UoM = poH + UpXyH = wH therefore w= Uy + UpXy
Therefore
‘= u,(relative pemeability) = 1+ xy 7.10
Ho
By rearranging the terms we get:
Xm — Br 1 7.11
In Table 7.1. the magnetic susceptibilities of paramagnetic and diamagnetic
materials are given.
From Figure 7.8 it is evident Ferromagnets have high positive susceptibility
right up to Curie Temperature. Only after Curie Temperature susceptibility
drastically falls. Where as Paramagnetic material has a graded response. At
very low temperatures susceptibility is high and it gradually falls with rise
in temperature.
Section 7.6.1. Classifications of Ferromagnets.
Feromagnets have further sub-classes namely Ferrimagnetic Materials and
Anti-ferromagnetic Materials.
In Figure 7.9 Chromium BCC crystal’s Unit Cell is shown. The Body
central left directed dipole moment is negated by the right directed dipole
moment of 8 corner atoms. This negation results into zero Magnetic
Moment because each of the 8 corner atoms make 1/8 contribution to the
Unit Cell.
Manganese Oxide is anti-ferromagnetic below Neel Temperature and above
Neel Temperature it is Paramagnetic. In Figure 7.10, it is shown below Neel
‘Temperature.
In 1986, high temperature Lanthanite Barium Copper Oxide [La(2-
x)Ba(x)CuO4] were found to exhibit superconductivity at 30K shown in
Figure 7.11. Yittrium Barium Copper Oxide(YBa2Cu306) Superconductors
were discovered which exhibited superconductivity at 90K warmer than
liquid Nitrogen Temperature. In all these layered compounds it was found
that CuO2 planes had anti-ferromagnetic order. This led to a flurry of
activities in CuO2 based Ceramics for developing Room Temperature
Superconductors. A detailed account of the quest for Room Temperature
Superconductor is given in Chapter 8.
In Figure 7.12 the dipole alignments in Ferro, Ferri and Antiferro-magnetic
materials are shown. Here Tc refers to Curie Temperature and Ty refers to
Neel Temperature.
Curie Temperature refers to Ferromagnetic Materials only. Above Curie
Temperature, Ferromagnetic Materials loses its spontaneous magnetization.
In Ferrimagnetic and Anti-Ferromagnetic materials below Neel
Temperature, the materials retain their Ferrimagnetism or
Antiferromagnetism as the case may be. But above Neel Temperature they
become paramagnetic. Thermal fluctuations disturbs the order.
Table 7.2. gives the Neel Temperature and Curie Temperature for some
important magnetic materials.
Table 7.2.Neel Temperature for some important Ferri and Anti-
Ferromagnetic Materials and Curie Temperatures for Ferromagnetic
Materials.
FERRI & ANTI- NEEL CURIE
FERRO TEMP. FERRO TEMP.
MnO 116K Fe 1043K
Cr 308K Co 1400K
MntTe 307K Ni 631
CoO 291K Gadolenium 292
FERRI & ANTI- NEEL CURIE
FERRO TEMP. FERRO TEMP.
NiO 925K MnBi 630
Fe203 948
FeOFe203 858
As is evident from the Table 7.14 , Fe, Co and Ni which are the transition
elements and which have uncompensated spin electrons are strongly
Ferromagnetic Materials
Section 7.6.2. Domains and Hysteresis.
Like Polycrystals or like Ferrielectric materials, in ferromagnetic and
ferrimagnetic materials also there are domains of magnetization. Each
domain has its own alignment direction and all dipoles in each domain are
aligned but the direction of alignment changes from domain to domain as
shown in Figure 7.15.
As we move from one domain to another, orientation of the dipoles only
gradually change. Thus there is no abrupt changes across the domain walls
as shown in Figure 7.16.
When external magnetic field is applied, favourable domains grow at the
expense of unfavourable domains as shown in Figure 7.17. Finally at high
fields a domain aligned with external field remains. This is an exact analog
of the electric poling of Ferro-electric material.
Figure 7.18 gives B-H curve of a ferro-magnet. Red is the Hysteresis Loop.
Blue is the initial magnetization. B, is the remnance Magnetic Flux Density
at H=0. Hc is the coercive magnetic field required to completely
demagnetize the material.
As is evident from figure 7.19 induced magnetization is perfectly linear in
paramagnetic and in diamagnetic materials but is non-linear in
ferro/ferromagnetic materials. Hence latter has a hysteresis loop.
Figure 7.3. Right Hand Palm Rule. Right hand fingers
clasp the solenoid in the direction of the current flow
and right thumb indicates the direction of magnetic
flux.
M, isthe
remnent Magnetic
Moment
Curie temperature
Figure 7.4 In Ferromagnetic materials dipole moments are aligned upto
1000K.Only above Curie Temperature that alignment get disrupted.
Figure 7.5., In paramagnetic materials an external field has to be applied to generate a small
magnetic field at Room Temperature. Room Temperature does not allow alignment of dipole
moments.
OO00
OO00
6@ee0
6@e60
6060
OOOO
66060
&606@
(b)
Figure 7.6. Comparative study of Diamagnetic and Paramagnetic materials
Paramagnetic
Flux density, B
Diamagnetic
Magnetic field strength, H
Figure 7.7. Linear Magnetization Curves for Paramagnetic Materials(rel.permeability >1),
Vacuum (rel.permeability = 1) and diamagnetic Materials(rel. permeability < 1).
+—— paramagnet +—_—_——— ferromagnet
I T
T curie
Figure 7.8. Susceptibilty vs Temperature for Paramagnet and for Ferromagnet
Table 7.1 Room-Temperature Magnetic Susceptibilities for Diamagnetic and
Paramagnetic Materials
Diamagnetics Paramagnetics
Susceptibility Susceptibility
Xm (volume) Xm (volume)
Material (SI units) Material (SI units)
Aluminum oxide —1.81 x 1075 Aluminum 2.07 x 107-5
Copper —0.96 x 10-° Chromium B13 107
Gold —3.44 x 1075 Chromium chloride 1.51 x 10°
Mercury —2.85 x 10° Manganese sulfate 3.70 x 10-5
Silicon —0.41 x 10-5 Molybdenum 1.19 x 10-4
Silver —2.38 x 10-5 Sodium 8.48 x 10~°
Sodium chloride —1.41 x 107° Titanium 1.81 x 1074
Zinc —1.56 x 107° Zirconium 1.09 x 1074
Figure 7.9. Anti-Ferrpmagnetic Material.
oO 6 6O
Mn+ wn
Figure 7.10.Anti parallel alignment of spin
magnetic moments for Manganese Oxide
(MnO). At low temperature , it is anti-
ferromagnetic but above Neel Temperature it
becomes paramagnetic.
Figure 7.11. Crystal structure of La2CuO4. The
CuOQ2 planes exhibit anti-ferromagnetic order.
Red balls are Cu(2+)
Blue balls are O(2-)
Green balls are La(3+).
[et
ferro- ferri- antiferro-
magnetism magnetism magnetism
for T< To for T < Ty
ferromagnetism ferri- and antiferro-
requires free electrons magnetism can also occur
in the conduction band in insulators
metals and
metals
metal-oxides
Figure 7.12. Illustration of magnetic dipole alignments in
Ferrmagnetic materials, in Ferrimagnetic materials and antiferro-
magnetic materials.
(Octahedral) (Octahedral) (Tetrahedral)
Figure 7.13. Introduction of Magnetic lons transforms a pure
Ferromagnetic into Ferrimagnetic material. Above Curie
Temperature it becomes Paramagnetic.
Table-Figure 7.14.Magnetic Properties are determined by the position of the Element in the Periodic Table.
<_-!1-ceeeec=e-ee = FS |S |
h
A
h
A
h
A
A
h
A
A
One an | Another domain
Domain wall
Figure 7.15.Domains in Ferromagnetic/Ferrimagnetic
materials.Within each domain all the dipoles are aligned
but direction of alignment changes from domain to
domain.
Mee
Domain
wall
Figure 7.16. Gradual change in dipole orientation
across a domain wall.
B, (M,)
Flux density, B (or magnetization, M)
U Magnetic field strength, H
Figure 7.17. Growth of a principal domain at the expense of other
domains during induced magnetization by externally applied field. This
H=6 is an exact analog to domain growth in Ferri-Electric Material during
= Electric Poling.
h Field removal or
reversal
Initial
magnetization
Figure 7.18. The Hysteresis Loop in a Ferromagnetic Material.
Ferromagnetic and
ferrimagnetic
Flux density, B (tesla)
Diamagnetic and
paramagnetic
0 25 50
Magnetic field strength, H (A/m)
Figure 7.19. 6-H curve for Ferromagnetic, Paramagnetic and
diamagnetic materials.
Chapter 7. Section 7.6.3. Hysteresis Loops of Hard Iron and Soft Iron.
Chapter 7, Section 7.6.3 to 7.10, give the theoretical basis of spontaneous
magnetization in Ferromagnetic Materials, describes Magneto Crystalline
Anisotropy,describes magneto-striction and describes Giant Magnet
Resistance used as hard disc memories in Computers.
Chapter 7. Section 7.6.3. Hysteresis Loops of Hard Iron and Soft Iron.
Figures are at the end of the module
In Figure 7.18. given in the previous module, the magnetization curves of
Ferromagnetic materials, Paramagnetic Materials and Diamagnetic
Materials are given. Para or Diamagnetic materials have linear
magnetization curve hence there are no hysteresis loops and Ferromagnetic
Materials have non-linear curves hence they have hysteresis loops.
Hysteresis loop implies hysteresis loss.
This means in magnetizing or demagnetizing a ferromagnetic material,
some energy is expended. The energy expended will depend on the ease
with which we can magnetize a ferromagnetic material.
Soft irons are used in electro-mechanical relay switches as shown in Figure
7.19. An actuating current easily magnetizes the soft iron core and attracts
the spring-loaded switch. In this process the electro-mechanical relay is
closed. As the current is turned off the switch is immediately opened by
spring action. The hysteresis loop of Soft Iron along side the hysteresis loop
Hard Iron is shown in Figure 7.20.
As seen in Figure 7.20, soft iron has a much higher saturation
Magnetization (Bs>) but hard iron has a much higher remnant
Magnetization (B,) as well as a much higher Coercive Magnetic Field(Hc).
Soft iron gives rise to temporary magnet whereas hard iron gives a
permanent magnet. Because of high remnant Magnetization as well as a
much higher Coercive Magnetic Field that hard iron becomes a permanent
magnet.
Section 7.7. Theoretical basis of spontaneous Magnetization in
Ferromagnets.
The theoretical basis of magnetism is the Orbital Angular Momentum of the
orbital electrons in an atom and the Spin Angular Momentum of the
spinning electron circulating around the nucleus of the atom.
Just as our Earth has orbital period of 365.25 days around the Sun but it has
a spin period of 24 hours around it spin axis. In a similar fashion orbital
electrons have orbital angular momentum J; as well as it has Js spin angular
momentum.
Both these angular momentums give rise to dipole moments. Their parallel
alignment can give rise to strong spontaneous magnetization as we find in
magnetic materials such as FERRO-MAGNETIC Materials.
Their random alignment can give rise to weak spontaneous magnetization
as we find in PARAMAGNETIC materials. Diamagnetic materials have no
magnetization.
In 1915, Albert Einstein in team with W.J. de Hass (the son-in-law of the
great Dutch physicist H.A.Lorentz) demonstrated that magnetism was a
result of the alignments of electron’s orbital magnetic moment and spin
magnetic moment. As shown in Figure 7.21. they attached a soft iron
cylinder from a friction less pivot. The soft iron was surrounded by
solenoid. Whenever a impulsive current was passed through the solenoid,
the soft iron got magnetized and experienced a rotary motion so as to keep
the overall Angular momentum equal to zero.
This amply demonstrated that ferromagnetism is a result of the alignment of
magnetic dipole moments due to orbital angular momentum and due to spin
angular momentum.
But as we will see shortly that the spontaneous magnetization is primarily
due to the spin angular momentum.
Section 7.7.1. Electron’s orbital magnetic moment and spin magnetic
moment.
By definition, a loop of orbiting electron has a dipole moment:
u = Loop current X loop cross — sectional area=1Xmr* (A.m*) yf |
v
I = loop current due to circulating electron = —q X ao 7.2
mr
L(orbital angular momentum of electron) = Moment of Inertia X Angular Velocity
Therefore
v
L=m,r* xX ~ = m,vr 7.3
Substituting (7.2) and (7.3) into ee 1)
up = —qX——Xar? =-2 = 4 .L 7.4
2ur
Magnetic Moment(p!) of Orbiting Electron is anti-parallel to Orbital
Angular Momentum (L) as shown in Figure 7.22..
Similarly:
Us = — = -S_ where S = Spin Angular Momentum re
m
In a preferred direction say Z-direction we have the projection of L and S
on Z-axis.
We have seen in Quantum Mechanics that :
h
Myz = ——" mh =! where m, = 0,+1,+2........+l where 1=(n—1) 7.6
2m, 4mm,
2. ee 75
Hsz — 2m, £0) — a e-¢h ;) =e
Here:
h eV
uz(Bohr Magneton) = a = 9.274 x 10774 Pe 5.788 x 10°>§ — 7.6.
4mm T r
Both spin and orbital angular momentum have a role to play as shown in
Figure 7.23.
After detailed investigation it was found that in ferro-magnetic materials,
spin angular momentum rather than orbital is the main contributor to Ferro-
magnetism. The orbital angular momentum have a role to play but when
there are uncompensated spins as in the case of transition elements the
orbitals have negligible role to play.
Ferromagnetism occurs because of coupling of uncompensated spins in
parallel direction. This coupling occurs directly and is called DIRECT
EXCHANGE COUPLING or through intermediate anions usually Oxygen
molecule through SUPER EXCHANGE.
In crystals this results in a net magnetic moments even at 300K. This is
purely a Pauli-Exclusion Phenomena and Coulombic Interaction
phenomena.
As shown in Figure 7.24., uncompensated spins of two atoms in an
overlapping electron clouds have preference for parallel alignment (which
contributes to net magnetic moment) rather than anti-parallel alignment
(which is zero magnetic moment). Parallel alignment corresponds to lower
energy level E> because of less columbic repulsion and anti-parallel
alignment corresponds to higher energy level E, because of stronger
columbic repulsion due to closer spatial proximity. So obviously the lower
energy state is preferred hence there is spontaneous magnetization in
elements with uncompensated spin electrons. This is the case for Transition
Elements hence Fe,Co and Ni are the strongest ferromagnetic materials.
AE = E, — E, = exchange energy 7.7
At room temperature:
AE > k,300K
Hence spontaneous magnetization is high up to Curie Temperature. At
Curie Temperature exchange coupling is disrupted by thermal fluctuations
and material becomes paramagnetic.
In Table 7.7.1. the Curie Temperatures of important Ferro-magnets are
listed.
Table 7.7.1. Curie Temperatures of typical Ferromagnetic materials.
Materials Curie Temperature (K)
Fe 1043
Co 1388
Ni 627
Gd 293
Section 7.8. Magneto-crystalline Anisotropy.
In Figure 7.25. M-H curve for a single iron crystal is given. M-H depends
on the crystal direction. As seen from the Figure 7.25, it is the easiest along
[100] direction and hardest along [111] direction.
Section 7.9. Magnetostriction.
In Figure 7.26. magnetostriction is defined. The Iron crystal elongates along
the easy X-direction but contracts along the Y-direction.
Section 7.10. Giant Magneto-Resistance used in hard discs of
Computers.
Giant Magneto-Resistance(GMR) is widely used in hard disc memories of
computers. It made its mass market debut when IBM commercialized its
record breaking 16.8GB hard disc in computer market. IBM called it SPIN
VALVE based on electronic spin.
In 1980, Peter Gruenberg of KFA Research Institute in Julich, Germany,
and Albert Fert of the University of Paris-Sud saw large resistance change
of 6% and 50% in Spin Valves.
In IBM, using sputtering,.scientists built trilayer GMR as shown in Figure
7.27. and demonstrated a large resistance change.
As shown in the Figure 7.27 there are two Ferromagnetic Layers of Co
separated by non-magnetic Cu layer.
A current flow through GMR experiences a Spin Valve effect.
What does this mean?
The tri-layer can have its ferromagnetic layer anti-parallel or parallel.
Anti-parallel FM layers behaves like a open valve offering large resistance
and parallel FM layers behave like a close valve offering small resistance.
This is better clarified by the Figure 7.28.
The scattering of electron depends on the spin of the conducting electron.
There are two cases:
Case 1: conducting electron spin is the same as the spin of the FM layer.
This will undergo very weak scattering. Hence low resistance.
Case 2: conducting electron spin is opposite the spin of the FM layer. This
will undergo very strong scattering Hence high resistance.
Now you examine the Figure 7.28.
We have two cases: Left Hand is parallel FM and right is anti-parallel FM.
Its equivalent electrical circuit, considering all permutation of spins, are
given below.
It is evident that Parallel FM has a much lower resistance and antiparallel
FM offers a very high resistance path. We study GMR in the case when the
current flows in the direction perpendicular to the layers. The GMR effect is
exploited in magnetic field sensors and its applications range from
automotive to information storage technology.
In Figure 7.29. the principle of longitudinal recording is illustrated.
B(Tesla=Weberim“2) Soft Magnetic Materials
Bs2
Bs1
Br
Hard Magnetic Materials
H(AT/m)
-Bs1
-Bs2
Figure 7.20. Hysteresis curves of soft
magnetic material and hard magnetic
material.
°?
SSS
Saar angular
{5 L momentum ?
bi = H & 1
a = af
a
s
B=0 BT
Figure 7.21. Experimental set-up for demonstrating
Einstein-de Hass effect.
Unit Vector normal to the plane of the loop
a
u
I
Lin
I
Zz
Figure 7.22. Magnetic dipole Moment due to a loop of
current! and loop cross-sectional area A
Mysterious : Spin moment Intuitive : Orbital moment
Figure 7.23. Contribution to masgnetic dipole moment due to spin on the left and
due to orbital motion on the right.
E1
E2
anti-parallel spin pair
Pauli Exclusion Principle allows
anti-parallel spins to be in close spatial
proximity but leading to stronger
Coulombic Repulsion hence
corresponds to higher energy state E1.
AtomA AtomB
parallel spin pair
Pauli Exclusion Principle does not allow
IN close spatial proximity between parallel
spin electrons. Infact parallel spin
experience Pauli Exclusion Spatial
Repulsion but in the process it
experiences less Coulombic Repulsion
AtomA AtomB _ hence it corresponds to lower Energy
Level E2.
E1-E2=Exchange Energy
Figure 7.24 Illustration of Exchange Energy due to
overlapping orbitals of two adjacent Atoms A and B.
Magnetizing field H (x10* A m'!)
0 l 2 3 4
2
15
< [111]
S Hard
x | [111]
5 Medium
g [110]
5
Ep 9.5
a Easy
[100] 4
0)
0 0.01 0.02 0.03 0.04 0.05 0.06
Applied magnetic field u/7 (T)
Figure 7.25. Magnetocrystalline anisotropy in single iron crystal
Original Fe crystal
<< >
sstcasaansutsinane y [010]
Lon
F N F
ms
R
(a) () |
Figure 7.27. The structure of multi-layerd Giant Magneto-Resistance (GMR) used in hard
discs of a computer.
.
TTTITIIT Ifill ifiiiitiiiiiiiiiiia
<§_— f+ 6f—>
Figure 7.26. Illustration of Magnetostriction.
Magnetic (ferromagnetic) metal
Nonmagnetic metal (spacer)
Magnetic metal
+
t
;
t
Ri R, RR
Figure 7.28. Parallel FM layers offer low resitance whereas
antiparallel FM layers offer high resistance path.
Input signal i(d)
Output signal
Fringe field
Record (write) Storage Read (play)
Figure 7.29.The principle of longitudinal magnetic recording on a flexible medium, e.g. magnetic tape
in an audio cassette
Chapter 8. Superconductors.
Chapter 8 gives the history and recent developments in the field of
Superconductors.
Chapter 8. Superconductors.[curtsey:
[Figures are at the end of the Module]
Superconductors, materials that have no resistance to the flow of
electricity, are one of the last great frontiers of scientific discovery. Not only
have the limits of superconductivity not yet been reached, but the theories
that explain superconductor behavior seem to be constantly under review. In
1911 superconductivity was first observed in mercury by Dutch physicist
Heike Kamerlingh Onnes of Leiden University. When he cooled it to the
temperature of liquid helium, 4 degrees Kelvin (-452F, -269C), its
resistance suddenly disappeared. The Kelvin scale represents an "absolute"
scale of temperature. Thus, it was necessary for Onnes to come within 4
degrees of the coldest temperature that is theoretically attainable to witness
the phenomenon of superconductivity. Later, in 1913, he won a Nobel Prize
in physics for his research in this area.
The next great milestone in understanding how matter behaves at extreme
cold temperatures occurred in 1933. German researchers Walther Meissner
(above left) and Robert Ochsenfeld (above right) discovered that a
superconducting material will repel a magnetic field (below graphic). A
magnet moving by a conductor induces currents in the conductor. This is
the principle on which the electric generator operates. But, in a
superconductor the induced currents exactly mirror the field that would
have otherwise penetrated the superconducting material - causing the
magnet to be repulsed. This phenomenon is known as strong diamagnetism
and is today often referred to as the "Meissner effect" (an eponym). The
Meissner effect is so strong that a magnet can actually be levitated over a
superconductive material.
In subsequent decades other superconducting metals, alloys and
compounds were discovered. In 1941 niobium-nitride was found to
superconduct at 16 K. In 1953 vanadium-silicon displayed superconductive
properties at 17.5 K. And, in 1962 scientists at Westinghouse developed the
first commercial superconducting wire, an alloy of niobium and titanium
(NbTi). High-energy, particle-accelerator electromagnets made of copper-
clad niobium-titanium were then developed in the 1960s at the Rutherford-
Appleton Laboratory in the UK, and were first employed in a
superconducting accelerator at the Fermilab Tevatron in the US in 1987.
The first widely-accepted theoretical understanding of superconductivity
was advanced in 1957 by American physicists John Bardeen, Leon Cooper,
and John Schrieffer (above). Their Theories ofSuperconductivity became
known as the BCS theory - derived from the first letter of each man's last
name - and won them a Nobel prize in 1972. The mathematically-complex
BCS theory explained superconductivity at temperatures close to absolute
zero for elements and simple alloys. However, at higher temperatures and
with different superconductor systems, the BCS theory has subsequently
become inadequate to fully explain how superconductivity is occurring.
Another significant theoretical advancement came in 1962 when Brian D.
Josephson, a graduate student at Cambridge University, predicted that
electrical current would flow between 2 superconducting materials - even
when they are separated by a non-superconductor or insulator. His
prediction was later confirmed and won him a share of the 1973 Nobel
Prize in Physics. This tunneling phenomenon is today known as the
"Josephson effect" and has been applied to electronic devices such as the
SQUID, an instrument capabable of detecting even the weakest magnetic
fields.
The 1980's were a decade of unrivaled discovery in the field of
superconductivity. In 1964 Bill Little of Stanford University had suggested
the possibility of organic (carbon-based) superconductors. The first of these
theoretical superconductors was successfully synthesized in 1980 by Danish
researcher Klaus Bechgaard of the University of Copenhagen and 3 French
team members. (TMTSF) PF¢ had to be cooled to an incredibly cold 1.2K
transition temperature (known as Tc) and subjected to high pressure to
superconduct. But, its mere existence proved the possibility of "designer"
molecules - molecules fashioned to perform in a predictable way.
Then, in 1986, a truly breakthrough discovery was made in the field of
superconductivity. Alex Miiller and Georg Bednorz , researchers at the IBM
Research Laboratory in Riischlikon, Switzerland, created a brittle ceramic
compound that superconducted at the highest temperature then known: 30
K. What made this discovery so remarkable was that ceramics are normally
insulators. They don't conduct electricity well at all. So, researchers had not
considered them as possible high-temperature superconductor candidates.
The Lanthanum, Barium, Copper and Oxygen compound that Miiller and
Bednorz synthesized, behaved in a not-as-yet-understood way. (Original
discovery of this first of the superconducting copper-oxides (cuprates) won
the 2 men a Nobel Prize the following year. It was later found that tiny
amounts of this material were actually superconducting at 58 K, due to a
small amount of lead having been added as a calibration standard - making
the discovery even more noteworthy.
Miller and Bednorz' discovery triggered a flurry of activity in the field
of superconductivity. Researchers around the world began "cooking" up
ceramics of every imaginable combination in a quest for higher and higher
Tc's. In January of 1987 a research team at the University of Alabama-
Huntsville substituted Yttrium for Lanthanum in the Miiller and Bednorz
molecule and achieved an incredible 92 K Tc. For the first time a material
(today referred to as YBCO) had been found that would superconduct at
temperatures warmer than liquid nitrogen - a commonly available coolant.
Additional milestones have since been achieved using exotic - and often
toxic - elements in the base perovskite ceramic. The current class (or
"system") of ceramic superconductors with the highest transition
temperatures are the mercuric-cuprates. The first synthesis of one of these
compounds was achieved in 1993 at the University of Colorado and by the
team of A. Schilling, M. Cantoni, J. D. Guo, and H. R. Ott of Zurich,
Switzerland. The world record Tc of 138 K is now held by a thallium-
doped, mercuric-cuprate comprised of the elements Mercury, Thallium,
Barium, Calcium, Copper and Oxygen. The Tc of this ceramic
superconductor was confirmed by Dr. Ron Goldfarb at the National Institute
of Standards and Technology-Colorado in February of 1994. Under extreme
pressure its Tc can be coaxed up even higher - approximately 25 to 30
degrees more at 300,000 atmospheres.
The first company to capitalize on high-temperature superconductors was
Illinois Superconductor (today known as ISCO International), formed in
1989. This amalgam of government, private-industry and academic interests
introduced a depth sensor for medical equipment that was able to operate at
liquid nitrogen temperatures (~ 77K).
In recent years, many discoveries regarding the novel nature of
superconductivity have been made. In 1997 researchers found that at a
temperature very near absolute zero an alloy of gold and indium was both a
superconductor and a natural magnet. Conventional wisdom held that a
material with such properties could not exist! Since then, over a half-dozen
such compounds have been found. Recent years have also seen the
discovery of the first high-temperature superconductor that does NOT
contain any copper (2000), and the first all-metal perovskite superconductor
(2001).
Also in 2001 a material that had been sitting on laboratory shelves for
decades was found to be an extraordinary new superconductor. Japanese
researchers measured the transition temperature of magnesium diboride at
39 Kelvin - far above the highest Tc of any of the elemental or binary alloy
superconductors. While 39 K is still well below the Tc's of the "warm"
ceramic superconductors, subsequent refinements in the way MgB, is
fabricated have paved the way for its use in industrial applications.
Laboratory testing has found MgB, will outperform NbTi and Nb3Sn wires
in high magnetic field applications like MRI.
Though a theory to explain high-temperature superconductivity still
eludes modern science, clues occasionally appear that contribute to our
understanding of the exotic nature of this phenomenon. In 2005, for
example, Superconductors.ORG discovered that increasing the weight
ratios of alternating planes within the layered perovskites can often increase
Ic significantly. This has led to the discovery of more than 70 new high-
temperature superconductors, including a candidate for a new world record.
The most recent "family" of superconductors to be discovered is the
"pnictides". These iron-based superconductors were first observed by a
group of Japanese researchers in 2006. Like the high-Tc copper-oxides, the
exact mechanism that facilitates superconductivity in them is a mystery.
However, with Tc's over 50K, a great deal of excitement has resulted from
their discovery.
Researchers do agree on one thing: discovery in the field of
superconductivity is as much serendipity as it is science.
Section 8.1.Meisnner Effect in Superconductors.
As the temperature falls below a critical temperature there is phase change
and the metal under consideration becomes super-conductor as shown in
Figure 8.1. In normal metal no such sudden drop in resistance is observed
or measured.
In Figure 8.2. a long cylindrical YBCO bar is shown kept in an external
magnetic field. Below critical temperature, as the metal becomes
superconductor it becomes strongly diamagnetic. In presence of an external
magnetic field, strong eddy currents are set up within the body which in
accordance with Lenz’s law completely excludes the external magnetic field
from the interior of the body as shown in Figure 8.2.b. As the metal returns
to normal condition above the critical temperature, external magnetic field
passes through the body but causes no induced magnetization as shown in
Figure 8.2.a.
Table 8.1. tabulates some well known superconductors and some recently
discovered high temperature ceramics of cuparate variety. The record high
critical temperature to date is 153K. It also tabulates the critical magnetic
field above which superconductivity gets killed. For ceramic YBCO type
superconductors also there is a critical Magnetic Field Hc) and Hc; but it
has not been shown in the Table.
Section 8.2. BCS theory of Superconductivity.
In 1957, John Bardeen, Leon Cooper and John Schriffer proposed the
cooper pair theory. According to this theory electrons are Fermions and real
particles with Js (Spin Angular Momentum)= +(1/2)h hence susceptible to
photon and defect scattering resulting in resistive metals.
But as temperature falls, electrons couple to form virtual particles with Js =
Oh. These are Bosons which are virtual particles and hence not susceptible
to photon and defect scattering. This leads to sudden drop in resistance as
shown in Figure 8.1.
This theory came to be known as BCS Theory.
BCS Theory predicts isotope effect in the following manner:
M°*T. = Constant here M = molar mass of the isotope 8.1
This equation predicts that for lighter isotopes super-conductivity can be
maintained till higher temperatures.
BCS Theory predicts a critical Magnetic Field also. External Magnetic
Fields greater than Bc (the critical magnetic flux density) kills the
superconductivity phase of the given metal.
BCS theory is strictly for metals. It completely fails to explain the
superconductivity in Cuparate Ceramics.
Table 8.1 Critical Temperatures and Magnetic Fluxes for Selected
Superconducting Materials
Critical Temperature Critical Magnetic Flux
Material Tc (K) Density Bc (tesla)"
Elements?
Tungsten ().02 0.0001
Titanium 0.40 0.0056
Aluminum 1.18 0.0105
Tin 3.72 0.0305
Mercury (a) 4.15 0.0411
Lead 7.19 0.0803
Compounds and Alloys”
Nb-Ti alloy 10.2 12
Nb-Zr alloy 10.8 11
PbMo¢Ss 14.0 45
V;Ga 16.5 22
Nb;Sn 18.3 22
Nb;Al 18.9 32
Nb,Ge 23.0 30
Ceramic Compounds
YBaoC u 30, 92 _
BipSr>CaoCu;, Oj 110 —_—
Tl) BapCa>Cu; Oj 125 —
HgBaCa,Cu.Ox 153 —
* The critical magnetic flux density (j¢o/7c) for the elements was measured at
0 K. For alloys and compounds, the flux is taken as jio//¢2 (in teslas), measured
at 0 K.
® Source: Adapted with permission from Materials at Low Temperatures, R. P.
Reed and A. F. Clark (Editors), American Society for Metals, Metals Park,
OH, 1983.
Electrical resistivity
Superconductor
Normal metal
0 Tc
Temperature (K)
Figure 8.1 Electrical Resistivity vs Temp. curve shows a sharp drop in
resistance of a superconductor below Tc (critical temperature). In normal
metal no such critical temperature exists.
(a)
T> T. a &,
Figure 8.2. A superconductor bar is kept in an external magnetic field. When the
temperature of the bar is below Tc, the field lines are undisturbed and pass
normally through the metallic cylinder. But as soon as it is cooled below Tc, it
abruptly excludes the external field as shown in (b). This drastic exclusion is
because of strong eddy currents set up in the cylinder. These eddy currents in
accordance with LENZ's Law oppose the exrernal field and hence exclude it from
the interior.
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